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3470 lines
106 KiB
3470 lines
106 KiB
%* glpk02.tex *%
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\chapter{Basic API Routines}
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\section{General conventions}
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\subsection{Library header}
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All GLPK API data types and routines are defined in the header file
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\verb|glpk.h|. It should be included in all source files which use
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GLPK API, either directly or indirectly through some other header file
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as follows:
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\begin{verbatim}
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#include <glpk.h>
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\end{verbatim}
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\subsection{Error handling}
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If some GLPK API routine detects erroneous or incorrect data passed by
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the application program, it writes appropriate diagnostic messages to
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the terminal and then abnormally terminates the application program.
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In most practical cases this allows to simplify programming by avoiding
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numerous checks of return codes. Thus, in order to prevent crashing the
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application program should check all data, which are suspected to be
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incorrect, before calling GLPK API routines.
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Should note that this kind of error handling is used only in cases of
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incorrect data passed by the application program. If, for example, the
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application program calls some GLPK API routine to read data from an
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input file and these data are incorrect, the GLPK API routine reports
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about error in the usual way by means of the return code.
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\subsection{Thread safety}
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The standard version of GLPK API is {\it not} thread safe and therefore
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should not be used in multi-threaded programs.
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\subsection{Array indexing}
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Normally all GLPK API routines start array indexing from 1, not from 0
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(except the specially stipulated cases). This means, for example, that
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if some vector $x$ of the length $n$ is passed as an array to some GLPK
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API routine, the latter expects vector components to be placed in
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locations \verb|x[1]|, \verb|x[2]|, \dots, \verb|x[n]|, and the
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location \verb|x[0]| normally is not used.
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To avoid indexing errors it is most convenient and most reliable to
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declare the array \verb|x| as follows:
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\begin{verbatim}
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double x[1+n];
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\end{verbatim}
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\noindent
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or to allocate it as follows:
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\begin{verbatim}
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double *x;
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. . .
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x = calloc(1+n, sizeof(double));
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. . .
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\end{verbatim}
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\noindent
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In both cases one extra location \verb|x[0]| is reserved that allows
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passing the array to GLPK routines in a usual way.
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\section{Problem object}
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All GLPK API routines deal with so called {\it problem object}, which
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is a program object of type \verb|glp_prob| and intended to represent
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a particular LP or MIP instance.
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The type \verb|glp_prob| is a data structure declared in the header
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file \verb|glpk.h| as follows:
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\begin{verbatim}
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typedef struct glp_prob glp_prob;
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\end{verbatim}
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Problem objects (i.e. program objects of the \verb|glp_prob| type) are
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allocated and managed internally by the GLPK API routines. The
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application program should never use any members of the \verb|glp_prob|
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structure directly and should deal only with pointers to these objects
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(that is, \verb|glp_prob *| values).
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The problem object consists of the following segments:
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\vspace*{-8pt}
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\begin{itemize}\setlength{\itemsep}{0pt}
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\item problem segment,
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\item basis segment,
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\item interior-point segment, and
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\item MIP segment.
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\end{itemize}
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\subsection{Problem segment}
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The {\it problem segment} contains original LP/MIP data, which
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corresponds to the problem formulation (1.1)---(1.3) (see Section
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\ref{seclp}, page \pageref{seclp}). This segment includes the following
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components:
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\vspace*{-8pt}
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\begin{itemize}\setlength{\itemsep}{0pt}
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\item rows (auxiliary variables),
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\item columns (structural variables),
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\item objective function, and
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\item constraint matrix.
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\end{itemize}
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\vspace*{-7pt}
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Rows and columns have the same set of the following attributes:
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\vspace*{-7pt}
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\begin{itemize}\setlength{\itemsep}{0pt}
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\item ordinal number,
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\item symbolic name (1 up to 255 arbitrary graphic characters),
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\item type (free, lower bound, upper bound, double bound, fixed),
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\item numerical values of lower and upper bounds,
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\item scale factor.
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\end{itemize}
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\vspace*{-7pt}
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{\it Ordinal numbers} are intended for referencing rows and columns.
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Row ordinal numbers are integers $1, 2, \dots, m$, and column ordinal
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numbers are integers $1, 2, \dots, n$, where $m$ and $n$ are,
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respectively, the current number of rows and columns in the problem
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object.
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{\it Symbolic names} are intended for informational purposes. They also
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can be used for referencing rows and columns.
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{\it Types and bounds} of rows (auxiliary variables) and columns
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(structural variables) are explained above (see Section \ref{seclp},
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page \pageref{seclp}).
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{\it Scale factors} are used internally for scaling rows and columns of
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the constraint matrix.
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Information about the {\it objective function} includes numerical
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values of objective coefficients and a flag, which defines the
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optimization direction (i.e. minimization or maximization).
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The {\it constraint matrix} is a $m \times n$ rectangular matrix built
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of constraint coefficients $a_{ij}$, which defines the system of linear
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constraints (1.2) (see Section \ref{seclp}, page \pageref{seclp}). This
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matrix is stored in the problem object in both row-wise and column-wise
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sparse formats.
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Once the problem object has been created, the application program can
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access and modify any components of the problem segment in arbitrary
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order.
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\subsection{Basis segment}
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The {\it basis segment} of the problem object keeps information related
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to the current basic solution. It includes:
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\vspace*{-8pt}
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\begin{itemize}\setlength{\itemsep}{0pt}
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\item row and column statuses,
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\item basic solution statuses,
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\item factorization of the current basis matrix, and
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\item basic solution components.
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\end{itemize}
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\vspace*{-8pt}
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The {\it row and column statuses} define which rows and columns are
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basic and which are non-basic. These statuses may be assigned either by
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the application program of by some API routines. Note that these
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statuses are always defined independently on whether the corresponding
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basis is valid or not.
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The {\it basic solution statuses} include the {\it primal status} and
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the {\it dual status}, which are set by the simplex-based solver once
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the problem has been solved. The primal status shows whether a primal
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basic solution is feasible, infeasible, or undefined. The dual status
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shows the same for a dual basic solution.
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The {\it factorization of the basis matrix} is some factorized form
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(like {\it LU}-factorization) of the current basis matrix (defined by
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the current row and column statuses). The factorization is used by
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simplex-based solvers and kept when the solver terminates the search.
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This feature allows efficiently reoptimizing the problem after some
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modifications (for example, after changing some bounds or objective
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coefficients). It also allows performing the post-optimal analysis (for
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example, computing components of the simplex table, etc.).
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The {\it basic solution components} include primal and dual values of
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all auxiliary and structural variables for the most recently obtained
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basic solution.
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\subsection{Interior-point segment}
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The {\it interior-point segment} contains interior-point solution
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components, which include the solution status, and primal and dual
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values of all auxiliary and structural variables.
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\subsection{MIP segment}
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The {\it MIP segment} is used only for MIP problems. This segment
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includes:
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\vspace*{-8pt}
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\begin{itemize}\setlength{\itemsep}{0pt}
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\item column kinds,
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\item MIP solution status, and
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\item MIP solution components.
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\end{itemize}
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\vspace*{-8pt}
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The {\it column kinds} define which columns (i.e. structural variables)
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are integer and which are continuous.
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The {\it MIP solution status} is set by the MIP solver and shows whether
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a MIP solution is integer optimal, integer feasible (non-optimal), or
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undefined.
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The {\it MIP solution components} are computed by the MIP solver and
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include primal values of all auxiliary and structural variables for the
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most recently obtained MIP solution.
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Note that in case of MIP problem the basis segment corresponds to
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the optimal solution of LP relaxation, which is also available to the
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application program.
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Currently the search tree is not kept in the MIP segment, so if the
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search has been terminated, it cannot be continued.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newpage
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\section{Problem creating and modifying routines}
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\subsection{glp\_create\_prob --- create problem object}
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\synopsis
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\begin{verbatim}
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glp_prob *glp_create_prob(void);
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\end{verbatim}
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\description
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The routine \verb|glp_create_prob| creates a new problem object, which
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initially is ``empty'', i.e. has no rows and columns.
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\returns
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The routine returns a pointer to the created object, which should be
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used in any subsequent operations on this object.
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\subsection{glp\_set\_prob\_name --- assign (change) problem name}
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\synopsis
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\begin{verbatim}
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void glp_set_prob_name(glp_prob *P, const char *name);
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\end{verbatim}
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\description
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The routine \verb|glp_set_prob_name| assigns a given symbolic
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\verb|name| (1 up to 255 characters) to the specified problem object.
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If the parameter \verb|name| is \verb|NULL| or empty string, the
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routine erases an existing symbolic name of the problem object.
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\subsection{glp\_set\_obj\_name --- assign (change) objective function
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name}
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\synopsis
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\begin{verbatim}
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void glp_set_obj_name(glp_prob *P, const char *name);
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\end{verbatim}
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\description
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The routine \verb|glp_set_obj_name| assigns a given symbolic
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\verb|name| (1 up to 255 characters) to the objective function of the
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specified problem object.
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If the parameter \verb|name| is \verb|NULL| or empty string, the
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routine erases an existing symbolic name of the objective function.
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\newpage
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\subsection{glp\_set\_obj\_dir --- set (change) optimization direction
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flag}
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\synopsis
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\begin{verbatim}
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void glp_set_obj_dir(glp_prob *P, int dir);
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\end{verbatim}
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\description
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The routine \verb|glp_set_obj_dir| sets (changes) the optimization
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direction flag (i.e. ``sense'' of the objective function) as specified
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by the parameter \verb|dir|:
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\verb|GLP_MIN| means minimization;
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\verb|GLP_MAX| means maximization.
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\noindent
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(Note that by default the problem is minimization.)
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\subsection{glp\_add\_rows --- add new rows to problem object}
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\synopsis
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\begin{verbatim}
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int glp_add_rows(glp_prob *P, int nrs);
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\end{verbatim}
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\description
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The routine \verb|glp_add_rows| adds \verb|nrs| rows (constraints) to
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the specified problem object. New rows are always added to the end of
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the row list, so the ordinal numbers of existing rows are not changed.
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Being added each new row is initially free (unbounded) and has empty
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list of the constraint coefficients.
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Each new row becomes a non-active (non-binding) constraint, i.e. the
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corresponding auxiliary variable is marked as basic.
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If the basis factorization exists, adding row(s) invalidates it.
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\returns
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The routine \verb|glp_add_rows| returns the ordinal number of the first
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new row added to the problem object.
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\subsection{glp\_add\_cols --- add new columns to problem object}
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\synopsis
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\begin{verbatim}
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int glp_add_cols(glp_prob *P, int ncs);
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\end{verbatim}
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\description
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The routine \verb|glp_add_cols| adds \verb|ncs| columns (structural
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variables) to the specified problem object. New columns are always
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added to the end of the column list, so the ordinal numbers of existing
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columns are not changed.
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Being added each new column is initially fixed at zero and has empty
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list of the constraint coefficients.
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Each new column is marked as non-basic, i.e. zero value of the
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corresponding structural variable becomes an active (binding) bound.
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If the basis factorization exists, it remains valid.
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\returns
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The routine \verb|glp_add_cols| returns the ordinal number of the first
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new column added to the problem object.
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\subsection{glp\_set\_row\_name --- assign (change) row name}
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\synopsis
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\begin{verbatim}
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void glp_set_row_name(glp_prob *P, int i, const char *name);
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\end{verbatim}
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\description
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The routine \verb|glp_set_row_name| assigns a given symbolic
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\verb|name| (1 up to 255 characters) to \verb|i|-th row (auxiliary
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variable) of the specified problem object.
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If the parameter \verb|name| is \verb|NULL| or empty string, the
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routine erases an existing name of $i$-th row.
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\subsection{glp\_set\_col\_name --- assign (change) column name}
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\synopsis
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\begin{verbatim}
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void glp_set_col_name(glp_prob *P, int j, const char *name);
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\end{verbatim}
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\description
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The routine \verb|glp_set_col_name| assigns a given symbolic
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\verb|name| (1 up to 255 characters) to \verb|j|-th column (structural
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variable) of the specified problem object.
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If the parameter \verb|name| is \verb|NULL| or empty string, the
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routine erases an existing name of $j$-th column.
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\subsection{glp\_set\_row\_bnds --- set (change) row bounds}
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\synopsis
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{\tt void glp\_set\_row\_bnds(glp\_prob *P, int i, int type,
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double lb, double ub);}
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\description
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The routine \verb|glp_set_row_bnds| sets (changes) the type and bounds
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of \verb|i|-th row (auxiliary variable) of the specified problem
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object.
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The parameters \verb|type|, \verb|lb|, and \verb|ub| specify the type,
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lower bound, and upper bound, respectively, as follows:
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\begin{center}
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\begin{tabular}{cr@{}c@{}ll}
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Type & \multicolumn{3}{c}{Bounds} & Comment \\
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\hline
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\verb|GLP_FR| & $-\infty <$ &$\ x\ $& $< +\infty$
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& Free (unbounded) variable \\
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\verb|GLP_LO| & $lb \leq$ &$\ x\ $& $< +\infty$
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& Variable with lower bound \\
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\verb|GLP_UP| & $-\infty <$ &$\ x\ $& $\leq ub$
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& Variable with upper bound \\
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\verb|GLP_DB| & $lb \leq$ &$\ x\ $& $\leq ub$
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& Double-bounded variable \\
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\verb|GLP_FX| & $lb =$ &$\ x\ $& $= ub$
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& Fixed variable \\
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\end{tabular}
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\end{center}
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\noindent
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where $x$ is the auxiliary variable associated with $i$-th row.
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If the row has no lower bound, the parameter \verb|lb| is ignored. If
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the row has no upper bound, the parameter \verb|ub| is ignored. If the
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row is an equality constraint (i.e. the corresponding auxiliary
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variable is of fixed type), only the parameter \verb|lb| is used while
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the parameter \verb|ub| is ignored.
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Being added to the problem object each row is initially free, i.e. its
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type is \verb|GLP_FR|.
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\subsection{glp\_set\_col\_bnds --- set (change) column bounds}
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\synopsis
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{\tt void glp\_set\_col\_bnds(glp\_prob *P, int j, int type,
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double lb, double ub);}
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\description
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The routine \verb|glp_set_col_bnds| sets (changes) the type and bounds
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of \verb|j|-th column (structural variable) of the specified problem
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object.
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The parameters \verb|type|, \verb|lb|, and \verb|ub| specify the type,
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lower bound, and upper bound, respectively, as follows:
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\begin{center}
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\begin{tabular}{cr@{}c@{}ll}
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Type & \multicolumn{3}{c}{Bounds} & Comment \\
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\hline
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\verb|GLP_FR| & $-\infty <$ &$\ x\ $& $< +\infty$
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& Free (unbounded) variable \\
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\verb|GLP_LO| & $lb \leq$ &$\ x\ $& $< +\infty$
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& Variable with lower bound \\
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\verb|GLP_UP| & $-\infty <$ &$\ x\ $& $\leq ub$
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& Variable with upper bound \\
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\verb|GLP_DB| & $lb \leq$ &$\ x\ $& $\leq ub$
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& Double-bounded variable \\
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\verb|GLP_FX| & $lb =$ &$\ x\ $& $= ub$
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& Fixed variable \\
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\end{tabular}
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\end{center}
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\noindent
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where $x$ is the structural variable associated with $j$-th column.
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If the column has no lower bound, the parameter \verb|lb| is ignored.
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If the column has no upper bound, the parameter \verb|ub| is ignored.
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If the column is of fixed type, only the parameter \verb|lb| is used
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while the parameter \verb|ub| is ignored.
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Being added to the problem object each column is initially fixed at
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zero, i.e. its type is \verb|GLP_FX| and both bounds are 0.
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\newpage
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\subsection{glp\_set\_obj\_coef --- set (change) objective coefficient
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or constant term}
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\synopsis
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\begin{verbatim}
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void glp_set_obj_coef(glp_prob *P, int j, double coef);
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\end{verbatim}
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\description
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The routine \verb|glp_set_obj_coef| sets (changes) the objective
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coefficient at \verb|j|-th column (structural variable). A new value of
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the objective coefficient is specified by the parameter \verb|coef|.
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If the parameter \verb|j| is 0, the routine sets (changes) the constant
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term (``shift'') of the objective function.
|
|
|
|
\subsection{glp\_set\_mat\_row --- set (replace) row of the constraint
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_mat_row(glp_prob *P, int i, int len, const int ind[],
|
|
const double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_mat_row| stores (replaces) the contents of
|
|
\verb|i|-th row of the constraint matrix of the specified problem
|
|
object.
|
|
|
|
Column indices and numerical values of new row elements should be
|
|
placed in locations\linebreak \verb|ind[1]|, \dots, \verb|ind[len]| and
|
|
\verb|val[1]|, \dots, \verb|val[len]|, respectively, where
|
|
$0 \leq$ \verb|len| $\leq n$ is the new length of $i$-th row, $n$ is
|
|
the current number of columns in the problem object. Elements with
|
|
identical column indices are not allowed. Zero elements are allowed,
|
|
but they are not stored in the constraint matrix.
|
|
|
|
If the parameter \verb|len| is 0, the parameters \verb|ind| and/or
|
|
\verb|val| can be specified as \verb|NULL|.
|
|
|
|
\note
|
|
|
|
If the basis factorization exists and changing the row changes
|
|
coefficients at basic column(s), the factorization is invalidated.
|
|
|
|
\subsection{glp\_set\_mat\_col --- set (replace) column of the
|
|
constr\-aint matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_mat_col(glp_prob *P, int j, int len, const int ind[],
|
|
const double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_mat_col| stores (replaces) the contents of
|
|
\verb|j|-th column of the constraint matrix of the specified problem
|
|
object.
|
|
|
|
Row indices and numerical values of new column elements should be
|
|
placed in locations\linebreak \verb|ind[1]|, \dots, \verb|ind[len]| and
|
|
\verb|val[1]|, \dots, \verb|val[len]|, respectively, where
|
|
$0 \leq$ \verb|len| $\leq m$ is the new length of $j$-th column, $m$ is
|
|
the current number of rows in the problem object. Elements with
|
|
identical row indices are not allowed. Zero elements are allowed, but
|
|
they are not stored in the constraint matrix.
|
|
|
|
If the parameter \verb|len| is 0, the parameters \verb|ind| and/or
|
|
\verb|val| can be specified as \verb|NULL|.
|
|
|
|
\note
|
|
|
|
If the basis factorization exists, changing the column corresponding
|
|
to a basic structural variable invalidates it.
|
|
|
|
\subsection{glp\_load\_matrix --- load (replace) the whole constraint
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_load_matrix(glp_prob *P, int ne, const int ia[],
|
|
const int ja[], const double ar[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_load_matrix| loads the constraint matrix passed
|
|
in the arrays \verb|ia|, \verb|ja|, and \verb|ar| into the specified
|
|
problem object. Before loading the current contents of the constraint
|
|
matrix is destroyed.
|
|
|
|
Constraint coefficients (elements of the constraint matrix) should be
|
|
specified as triplets (\verb|ia[k]|, \verb|ja[k]|, \verb|ar[k]|) for
|
|
$k=1,\dots,ne$, where \verb|ia[k]| is the row index, \verb|ja[k]| is
|
|
the column index, and \verb|ar[k]| is a numeric value of corresponding
|
|
constraint coefficient. The parameter \verb|ne| specifies the total
|
|
number of (non-zero) elements in the matrix to be loaded. Coefficients
|
|
with identical indices are not allowed. Zero coefficients are allowed,
|
|
however, they are not stored in the constraint matrix.
|
|
|
|
If the parameter \verb|ne| is 0, the parameters \verb|ia|, \verb|ja|,
|
|
and/or \verb|ar| can be specified as \verb|NULL|.
|
|
|
|
\note
|
|
|
|
If the basis factorization exists, this operation invalidates it.
|
|
|
|
\subsection{glp\_check\_dup --- check for duplicate elements in sparse
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
{\tt int glp\_check\_dup(int m, int n, int ne, const int ia[],
|
|
const int ja[]);}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_check_dup checks| for duplicate elements (that
|
|
is, elements with identical indices) in a sparse matrix specified in
|
|
the coordinate format.
|
|
|
|
The parameters $m$ and $n$ specifies, respectively, the number of rows
|
|
and columns in the matrix, $m\geq 0$, $n\geq 0$.
|
|
|
|
The parameter {\it ne} specifies the number of (structurally) non-zero
|
|
elements in the matrix,\linebreak {\it ne} $\geq 0$.
|
|
|
|
Elements of the matrix are specified as doublets $(ia[k],ja[k])$ for
|
|
$k=1,\dots,ne$, where $ia[k]$ is a row index, $ja[k]$ is a column
|
|
index.
|
|
|
|
The routine \verb|glp_check_dup| can be used prior to a call to the
|
|
routine \verb|glp_load_matrix| to check that the constraint matrix to
|
|
be loaded has no duplicate elements.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0& the matrix representation is correct;\\
|
|
$-k$& indices $ia[k]$ or/and $ja[k]$ are out of range;\\
|
|
$+k$& element $(ia[k],ja[k])$ is duplicate.\\
|
|
\end{retlist}
|
|
|
|
\subsection{glp\_sort\_matrix --- sort elements of the constraint
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_sort_matrix(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_sort_matrix| sorts elements of the constraint
|
|
matrix by rebuilding its row and column linked lists.
|
|
|
|
On exit from the routine the constraint matrix is not changed, however,
|
|
elements in the row linked lists become ordered by ascending column
|
|
indices, and the elements in the column linked lists become ordered by
|
|
ascending row indices.
|
|
|
|
\subsection{glp\_del\_rows --- delete rows from problem object}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_del_rows(glp_prob *P, int nrs, const int num[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_del_rows| deletes rows from the specified problem
|
|
object. Ordinal numbers of rows to be deleted should be placed in
|
|
locations \verb|num[1]|, \dots, \verb|num[nrs]|, where ${\tt nrs}>0$.
|
|
|
|
Note that deleting rows involves changing ordinal numbers of other
|
|
rows remaining in the problem object. New ordinal numbers of the
|
|
remaining rows are assigned under the assumption that the original
|
|
order of rows is not changed. Let, for example, before deletion there
|
|
be five rows $a$, $b$, $c$, $d$, $e$ with ordinal numbers 1, 2, 3, 4,
|
|
5, and let rows $b$ and $d$ have been deleted. Then after deletion the
|
|
remaining rows $a$, $c$, $e$ are assigned new oridinal numbers 1, 2, 3.
|
|
|
|
If the basis factorization exists, deleting active (binding) rows,
|
|
i.e. whose auxiliary variables are marked as non-basic, invalidates it.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_del\_cols --- delete columns from problem object}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_del_cols(glp_prob *P, int ncs, const int num[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_del_cols| deletes columns from the specified
|
|
problem object. Ordinal numbers of columns to be deleted should be
|
|
placed in locations \verb|num[1]|, \dots, \verb|num[ncs]|, where
|
|
${\tt ncs}>0$.
|
|
|
|
Note that deleting columns involves changing ordinal numbers of other
|
|
columns remaining in\linebreak the problem object. New ordinal numbers
|
|
of the remaining columns are assigned under the assumption that the
|
|
original order of columns is not changed. Let, for example, before
|
|
deletion there be six columns $p$, $q$, $r$, $s$, $t$, $u$ with
|
|
ordinal numbers 1, 2, 3, 4, 5, 6, and let columns $p$, $q$, $s$ have
|
|
been deleted. Then after deletion the remaining columns $r$, $t$, $u$
|
|
are assigned new ordinal numbers 1, 2, 3.
|
|
|
|
If the basis factorization exists, deleting basic columns invalidates
|
|
it.
|
|
|
|
\subsection{glp\_copy\_prob --- copy problem object content}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_copy_prob(glp_prob *dest, glp_prob *prob, int names);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_copy_prob| copies the content of the problem
|
|
object \verb|prob| to the problem object \verb|dest|.
|
|
|
|
The parameter \verb|names| is a flag. If it is \verb|GLP_ON|,
|
|
the routine also copies all symbolic names; otherwise, if it is
|
|
\verb|GLP_OFF|, no symbolic names are copied.
|
|
|
|
\subsection{glp\_erase\_prob --- erase problem object content}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_erase_prob(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_erase_prob| erases the content of the specified
|
|
problem object. The effect of this operation is the same as if the
|
|
problem object would be deleted with the routine \verb|glp_delete_prob|
|
|
and then created anew with the routine \verb|glp_create_prob|, with the
|
|
only exception that the pointer to the problem object remains valid.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_delete\_prob --- delete problem object}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_delete_prob(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_delete_prob| deletes a problem object, which the
|
|
parameter \verb|lp| points to, freeing all the memory allocated to this
|
|
object.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Problem retrieving routines}
|
|
|
|
\subsection{glp\_get\_prob\_name --- retrieve problem name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
const char *glp_get_prob_name(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_prob_name| returns a pointer to an internal
|
|
buffer, which contains symbolic name of the problem. However, if the
|
|
problem has no assigned name, the routine returns \verb|NULL|.
|
|
|
|
\subsection{glp\_get\_obj\_name --- retrieve objective function name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
const char *glp_get_obj_name(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_obj_name| returns a pointer to an internal
|
|
buffer, which contains symbolic name assigned to the objective
|
|
function. However, if the objective function has no assigned name, the
|
|
routine returns \verb|NULL|.
|
|
|
|
\subsection{glp\_get\_obj\_dir --- retrieve optimization direction
|
|
flag}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_obj_dir(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_obj_dir| returns the optimization direction
|
|
flag (i.e. ``sense'' of the objective function):
|
|
|
|
\verb|GLP_MIN| means minimization;
|
|
|
|
\verb|GLP_MAX| means maximization.
|
|
|
|
\subsection{glp\_get\_num\_rows --- retrieve number of rows}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_num_rows(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_num_rows| returns the current number of rows
|
|
in the specified problem object.
|
|
|
|
\subsection{glp\_get\_num\_cols --- retrieve number of columns}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_num_cols(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_num_cols| returns the current number of
|
|
columns in the specified problem object.
|
|
|
|
\subsection{glp\_get\_row\_name --- retrieve row name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
const char *glp_get_row_name(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_name| returns a pointer to an internal
|
|
buffer, which contains a symbolic name assigned to \verb|i|-th row.
|
|
However, if the row has no assigned name, the routine returns
|
|
\verb|NULL|.
|
|
|
|
\subsection{glp\_get\_col\_name --- retrieve column name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
const char *glp_get_col_name(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_name| returns a pointer to an internal
|
|
buffer, which contains a symbolic name assigned to \verb|j|-th column.
|
|
However, if the column has no assigned name, the routine returns
|
|
\verb|NULL|.
|
|
|
|
\subsection{glp\_get\_row\_type --- retrieve row type}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_row_type(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_type| returns the type of \verb|i|-th
|
|
row, i.e. the type of corresponding auxiliary variable, as follows:
|
|
|
|
\verb|GLP_FR| --- free (unbounded) variable;
|
|
|
|
\verb|GLP_LO| --- variable with lower bound;
|
|
|
|
\verb|GLP_UP| --- variable with upper bound;
|
|
|
|
\verb|GLP_DB| --- double-bounded variable;
|
|
|
|
\verb|GLP_FX| --- fixed variable.
|
|
|
|
\subsection{glp\_get\_row\_lb --- retrieve row lower bound}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_row_lb(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_lb| returns the lower bound of
|
|
\verb|i|-th row, i.e. the lower bound of corresponding auxiliary
|
|
variable. However, if the row has no lower bound, the routine returns
|
|
\verb|-DBL_MAX|.
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\subsection{glp\_get\_row\_ub --- retrieve row upper bound}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_row_ub(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_ub| returns the upper bound of
|
|
\verb|i|-th row, i.e. the upper bound of corresponding auxiliary
|
|
variable. However, if the row has no upper bound, the routine returns
|
|
\verb|+DBL_MAX|.
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\subsection{glp\_get\_col\_type --- retrieve column type}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_col_type(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_type| returns the type of \verb|j|-th
|
|
column, i.e. the type of corresponding structural variable, as follows:
|
|
|
|
\verb|GLP_FR| --- free (unbounded) variable;
|
|
|
|
\verb|GLP_LO| --- variable with lower bound;
|
|
|
|
\verb|GLP_UP| --- variable with upper bound;
|
|
|
|
\verb|GLP_DB| --- double-bounded variable;
|
|
|
|
\verb|GLP_FX| --- fixed variable.
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\subsection{glp\_get\_col\_lb --- retrieve column lower bound}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_col_lb(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_lb| returns the lower bound of
|
|
\verb|j|-th column, i.e. the lower bound of corresponding structural
|
|
variable. However, if the column has no lower bound, the routine
|
|
returns \verb|-DBL_MAX|.
|
|
|
|
\subsection{glp\_get\_col\_ub --- retrieve column upper bound}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_col_ub(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_ub| returns the upper bound of
|
|
\verb|j|-th column, i.e. the upper bound of corresponding structural
|
|
variable. However, if the column has no upper bound, the routine
|
|
returns \verb|+DBL_MAX|.
|
|
|
|
\subsection{glp\_get\_obj\_coef --- retrieve objective coefficient or
|
|
constant term}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_obj_coef(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_obj_coef| returns the objective coefficient
|
|
at \verb|j|-th structural variable (column).
|
|
|
|
If the parameter \verb|j| is 0, the routine returns the constant term
|
|
(``shift'') of the objective function.
|
|
|
|
\subsection{glp\_get\_num\_nz --- retrieve number of constraint
|
|
coefficients}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_num_nz(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_num_nz| returns the number of non-zero
|
|
elements in the constraint matrix of the specified problem object.
|
|
|
|
\subsection{glp\_get\_mat\_row --- retrieve row of the constraint
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_mat_row(glp_prob *P, int i, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_get_mat_row| scans (non-zero) elements of
|
|
\verb|i|-th row of the constraint matrix of the specified problem
|
|
object and stores their column indices and numeric values to locations
|
|
\verb|ind[1]|, \dots, \verb|ind[len]| and \verb|val[1]|, \dots,
|
|
\verb|val[len]|, respectively, where $0\leq{\tt len}\leq n$ is the
|
|
number of elements in $i$-th row, $n$ is the number of columns.
|
|
|
|
The parameter \verb|ind| and/or \verb|val| can be specified as
|
|
\verb|NULL|, in which case corresponding information is not stored.
|
|
|
|
\newpage
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_mat_row| returns the length \verb|len|, i.e.
|
|
the number of (non-zero) elements in \verb|i|-th row.
|
|
|
|
\subsection{glp\_get\_mat\_col --- retrieve column of the constraint
|
|
matrix}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_mat_col(glp_prob *P, int j, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_get_mat_col| scans (non-zero) elements of
|
|
\verb|j|-th column of the constraint matrix of the specified problem
|
|
object and stores their row indices and numeric values to locations
|
|
\linebreak \verb|ind[1]|, \dots, \verb|ind[len]| and \verb|val[1]|,
|
|
\dots, \verb|val[len]|, respectively, where $0\leq{\tt len}\leq m$ is
|
|
the number of elements in $j$-th column, $m$ is the number of rows.
|
|
|
|
The parameter \verb|ind| and/or \verb|val| can be specified as
|
|
\verb|NULL|, in which case corresponding information is not stored.
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_mat_col| returns the length \verb|len|, i.e.
|
|
the number of (non-zero) elements in \verb|j|-th column.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Row and column searching routines}
|
|
|
|
Sometimes it may be necessary to find rows and/or columns by their
|
|
names (assigned with the routines \verb|glp_set_row_name| and
|
|
\verb|glp_set_col_name|). Though a particular row/column can be found
|
|
by its name using simple enumeration of all rows/columns, in case of
|
|
large instances such a {\it linear} search may take too long time.
|
|
|
|
To significantly reduce the search time the application program may
|
|
create the row/column name index, which is an auxiliary data structure
|
|
implementing a {\it binary} search. Even in worst cases the search
|
|
takes logarithmic time, i.e. the time needed to find a particular row
|
|
(or column) by its name is $O(\log_2m)$ (or $O(\log_2n)$), where $m$
|
|
and $n$ are, resp., the number of rows and columns in the problem
|
|
object.
|
|
|
|
It is important to note that:
|
|
|
|
1. On creating the problem object with the routine
|
|
\verb|glp_create_prob| the name index is {\it not} created.
|
|
|
|
2. The name index can be created (destroyed) at any time with the
|
|
routine \verb|glp_create_index| (\verb|glp_delete_index|). Having been
|
|
created the name index becomes part of the corresponding problem
|
|
object.
|
|
|
|
3. The time taken to create the name index is $O[(m+n)\log_2(m+n)]$,
|
|
so it is recommended to create the index only once, for example, just
|
|
after the problem object was created.
|
|
|
|
4. If the name index exists, it is automatically updated every time
|
|
the name of a row/column is assigned/changed. The update operation
|
|
takes logarithmic time.
|
|
|
|
5. If the name index does not exist, the application should not call
|
|
the routines \verb|glp_find_row| and \verb|glp_find_col|. Otherwise,
|
|
an error message will be issued and abnormal program termination will
|
|
occur.
|
|
|
|
6. On destroying the problem object with the routine
|
|
\verb|glp_delete_prob|, the name index, if exists, is automatically
|
|
destroyed.
|
|
|
|
\subsection{glp\_create\_index --- create the name index}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_create_index(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_create_index| creates the name index for the
|
|
specified problem object. The name index is an auxiliary data
|
|
structure, which is intended to quickly (i.e. for logarithmic time)
|
|
find rows and columns by their names.
|
|
|
|
This routine can be called at any time. If the name index already
|
|
exists, the routine does nothing.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_find\_row --- find row by its name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_find_row(glp_prob *P, const char *name);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_find_row| returns the ordinal number of a row,
|
|
which is assigned the specified symbolic \verb|name|. If no such row
|
|
exists, the routine returns 0.
|
|
|
|
\subsection{glp\_find\_col --- find column by its name}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_find_col(glp_prob *P, const char *name);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_find_col| returns the ordinal number of a column,
|
|
which is assigned the specified symbolic \verb|name|. If no such column
|
|
exists, the routine returns 0.
|
|
|
|
\subsection{glp\_delete\_index --- delete the name index}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_delete_index(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_delete_index| deletes the name index previously
|
|
created by the routine\linebreak \verb|glp_create_index| and frees the
|
|
memory allocated to this auxiliary data structure.
|
|
|
|
This routine can be called at any time. If the name index does not
|
|
exist, the routine does nothing.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Problem scaling routines}
|
|
|
|
\subsection{Background}
|
|
|
|
In GLPK the {\it scaling} means a linear transformation applied to the
|
|
constraint matrix to improve its numerical properties.\footnote{In many
|
|
cases a proper scaling allows making the constraint matrix to be better
|
|
conditioned, i.e. decreasing its condition number, that makes
|
|
computations numerically more stable.}
|
|
|
|
The main equality is the following:
|
|
$$\widetilde{A}=RAS,\eqno(2.1)$$
|
|
where $A=(a_{ij})$ is the original constraint matrix, $R=(r_{ii})>0$ is
|
|
a diagonal matrix used to scale rows (constraints), $S=(s_{jj})>0$ is a
|
|
diagonal matrix used to scale columns (variables), $\widetilde{A}$ is
|
|
the scaled constraint matrix.
|
|
|
|
From (2.1) it follows that in the {\it scaled} problem instance each
|
|
original constraint coefficient $a_{ij}$ is replaced by corresponding
|
|
scaled constraint coefficient:
|
|
$$\widetilde{a}_{ij}=r_{ii}a_{ij}s_{jj}.\eqno(2.2)$$
|
|
|
|
Note that the scaling is performed internally and therefore
|
|
transparently to the user. This means that on API level the user always
|
|
deal with unscaled data.
|
|
|
|
Scale factors $r_{ii}$ and $s_{jj}$ can be set or changed at any time
|
|
either directly by the application program in a problem specific way
|
|
(with the routines \verb|glp_set_rii| and \verb|glp_set_sjj|), or by
|
|
some API routines intended for automatic scaling.
|
|
|
|
\subsection{glp\_set\_rii --- set (change) row scale factor}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_rii(glp_prob *P, int i, double rii);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_rii| sets (changes) the scale factor $r_{ii}$
|
|
for $i$-th row of the specified problem object.
|
|
|
|
\subsection{glp\_set\_sjj --- set (change) column scale factor}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_sjj(glp_prob *P, int j, double sjj);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_sjj| sets (changes) the scale factor $s_{jj}$
|
|
for $j$-th column of the specified problem object.
|
|
|
|
\subsection{glp\_get\_rii --- retrieve row scale factor}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_rii(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_rii| returns current scale factor $r_{ii}$
|
|
for $i$-th row of the specified problem object.
|
|
|
|
\vspace*{-6pt}
|
|
|
|
\subsection{glp\_get\_sjj --- retrieve column scale factor}
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_sjj(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_sjj| returns current scale factor $s_{jj}$
|
|
for $j$-th column of the specified problem object.
|
|
|
|
\vspace*{-6pt}
|
|
|
|
\subsection{glp\_scale\_prob --- scale problem data}
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_scale_prob(glp_prob *P, int flags);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_scale_prob| performs automatic scaling of problem
|
|
data for the specified problem object.
|
|
|
|
The parameter \verb|flags| specifies scaling options used by the
|
|
routine. The options can be combined with the bitwise OR operator and
|
|
may be the following:
|
|
|
|
\verb|GLP_SF_GM | --- perform geometric mean scaling;
|
|
|
|
\verb|GLP_SF_EQ | --- perform equilibration scaling;
|
|
|
|
\verb|GLP_SF_2N | --- round scale factors to nearest power of two;
|
|
|
|
\verb|GLP_SF_SKIP| --- skip scaling, if the problem is well scaled.
|
|
|
|
The parameter \verb|flags| may be also specified as \verb|GLP_SF_AUTO|,
|
|
in which case the routine chooses the scaling options automatically.
|
|
|
|
\vspace*{-6pt}
|
|
|
|
\subsection{glp\_unscale\_prob --- unscale problem data}
|
|
|
|
\vspace*{-4pt}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_unscale_prob(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
The routine \verb|glp_unscale_prob| performs unscaling of problem data
|
|
for the specified problem object.
|
|
|
|
``Unscaling'' means replacing the current scaling matrices $R$ and $S$
|
|
by unity matrices that cancels the scaling effect.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{LP basis constructing routines}
|
|
|
|
\subsection{Background}
|
|
|
|
To start the search the simplex method needs a valid initial basis.
|
|
In GLPK the basis is completely defined by a set of {\it statuses}
|
|
assigned to {\it all} (auxiliary and structural) variables, where the
|
|
status may be one of the following:
|
|
|
|
\verb|GLP_BS| --- basic variable;
|
|
|
|
\verb|GLP_NL| --- non-basic variable having active lower bound;
|
|
|
|
\verb|GLP_NU| --- non-basic variable having active upper bound;
|
|
|
|
\verb|GLP_NF| --- non-basic free variable;
|
|
|
|
\verb|GLP_NS| --- non-basic fixed variable.
|
|
|
|
The basis is {\it valid}, if the basis matrix, which is a matrix built
|
|
of columns of the augmented constraint matrix $(I\:|-A)$ corresponding
|
|
to basic variables, is non-singular. This, in particular, means that
|
|
the number of basic variables must be the same as the number of rows in
|
|
the problem object. (For more details see Section \ref{lpbasis}, page
|
|
\pageref{lpbasis}.)
|
|
|
|
Any initial basis may be constructed (or restored) with the API
|
|
routines \verb|glp_set_row_stat| and \verb|glp_set_col_stat| by
|
|
assigning appropriate statuses to auxiliary and structural variables.
|
|
Another way to construct an initial basis is to use API routines like
|
|
\verb|glp_adv_basis|, which implement so called
|
|
{\it crashing}.\footnote{This term is from early linear programming
|
|
systems and means a heuristic to construct a valid initial basis.} Note
|
|
that on normal exit the simplex solver remains the basis valid, so in
|
|
case of reoptimization there is no need to construct an initial basis
|
|
from scratch.
|
|
|
|
\subsection{glp\_set\_row\_stat --- set (change) row status}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_row_stat(glp_prob *P, int i, int stat);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_row_stat| sets (changes) the current status
|
|
of \verb|i|-th row (auxiliary variable) as specified by the parameter
|
|
\verb|stat|:
|
|
|
|
\verb|GLP_BS| --- make the row basic (make the constraint inactive);
|
|
|
|
\verb|GLP_NL| --- make the row non-basic (make the constraint active);
|
|
|
|
\verb|GLP_NU| --- make the row non-basic and set it to the upper bound;
|
|
if the row is not double-bounded, this status is equivalent to
|
|
\verb|GLP_NL| (only in case of this routine);
|
|
|
|
\verb|GLP_NF| --- the same as \verb|GLP_NL| (only in case of this
|
|
routine);
|
|
|
|
\verb|GLP_NS| --- the same as \verb|GLP_NL| (only in case of this
|
|
routine).
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_set\_col\_stat --- set (change) column status}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_col_stat(glp_prob *P, int j, int stat);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_col_stat sets| (changes) the current status
|
|
of \verb|j|-th column (structural variable) as specified by the
|
|
parameter \verb|stat|:
|
|
|
|
\verb|GLP_BS| --- make the column basic;
|
|
|
|
\verb|GLP_NL| --- make the column non-basic;
|
|
|
|
\verb|GLP_NU| --- make the column non-basic and set it to the upper
|
|
bound; if the column is not double-bounded, this status is equivalent
|
|
to \verb|GLP_NL| (only in case of this routine);
|
|
|
|
\verb|GLP_NF| --- the same as \verb|GLP_NL| (only in case of this
|
|
routine);
|
|
|
|
\verb|GLP_NS| --- the same as \verb|GLP_NL| (only in case of this
|
|
routine).
|
|
|
|
\subsection{glp\_std\_basis --- construct standard initial LP basis}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_std_basis(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_std_basis| constructs the ``standard'' (trivial)
|
|
initial LP basis for the specified problem object.
|
|
|
|
In the ``standard'' LP basis all auxiliary variables (rows) are basic,
|
|
and all structural variables (columns) are non-basic (so the
|
|
corresponding basis matrix is unity).
|
|
|
|
\subsection{glp\_adv\_basis --- construct advanced initial LP basis}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_adv_basis(glp_prob *P, int flags);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_adv_basis| constructs an advanced initial LP
|
|
basis for the specified problem object.
|
|
|
|
The parameter \verb|flags| is reserved for use in the future and must
|
|
be specified as zero.
|
|
|
|
In order to construct the advanced initial LP basis the routine does
|
|
the following:
|
|
|
|
1) includes in the basis all non-fixed auxiliary variables;
|
|
|
|
2) includes in the basis as many non-fixed structural variables as
|
|
possible keeping the triangular form of the basis matrix;
|
|
|
|
3) includes in the basis appropriate (fixed) auxiliary variables to
|
|
complete the basis.
|
|
|
|
As a result the initial LP basis has as few fixed variables as possible
|
|
and the corresponding basis matrix is triangular.
|
|
|
|
\subsection{glp\_cpx\_basis --- construct Bixby's initial LP basis}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_cpx_basis(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_cpx_basis| constructs an initial basis for the
|
|
specified problem object with the algorithm proposed by
|
|
R.~Bixby.\footnote{Robert E. Bixby, ``Implementing the Simplex Method:
|
|
The Initial Basis.'' ORSA Journal on Computing, Vol. 4, No. 3, 1992,
|
|
pp. 267-84.}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Simplex method routines}
|
|
|
|
The {\it simplex method} is a well known efficient numerical procedure
|
|
to solve LP problems.
|
|
|
|
On each iteration the simplex method transforms the original system of
|
|
equaility constraints (1.2) resolving them through different sets of
|
|
variables to an equivalent system called {\it the simplex table} (or
|
|
sometimes {\it the simplex tableau}), which has the following form:
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}r@{\:}c@{\:}r@{\:}c@{\:}r}
|
|
z&=&d_1(x_N)_1&+&d_2(x_N)_2&+ \dots +&d_n(x_N)_n \\
|
|
(x_B)_1&=&\xi_{11}(x_N)_1& +& \xi_{12}(x_N)_2& + \dots +&
|
|
\xi_{1n}(x_N)_n \\
|
|
(x_B)_2&=& \xi_{21}(x_N)_1& +& \xi_{22}(x_N)_2& + \dots +&
|
|
\xi_{2n}(x_N)_n \\
|
|
\multicolumn{7}{c}
|
|
{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .} \\
|
|
(x_B)_m&=&\xi_{m1}(x_N)_1& +& \xi_{m2}(x_N)_2& + \dots +&
|
|
\xi_{mn}(x_N)_n \\
|
|
\end{array} \eqno (2.3)
|
|
$$
|
|
where: $(x_B)_1, (x_B)_2, \dots, (x_B)_m$ are basic variables;
|
|
$(x_N)_1, (x_N)_2, \dots, (x_N)_n$ are non-basic variables;
|
|
$d_1, d_2, \dots, d_n$ are reduced costs;
|
|
$\xi_{11}, \xi_{12}, \dots, \xi_{mn}$ are coefficients of the
|
|
simplex table. (May note that the original LP problem (1.1)---(1.3)
|
|
also has the form of a simplex table, where all equalities are resolved
|
|
through auxiliary variables.)
|
|
|
|
From the linear programming theory it is known that if an optimal
|
|
solution of the LP problem (1.1)---(1.3) exists, it can always be
|
|
written in the form (2.3), where non-basic variables are set on their
|
|
bounds while values of the objective function and basic variables are
|
|
determined by the corresponding equalities of the simplex table.
|
|
|
|
A set of values of all basic and non-basic variables determined by the
|
|
simplex table is called {\it basic solution}. If all basic variables
|
|
are within their bounds, the basic solution is called {\it (primal)
|
|
feasible}, otherwise it is called {\it (primal) infeasible}. A feasible
|
|
basic solution, which provides a smallest (in case of minimization) or
|
|
a largest (in case of maximization) value of the objective function is
|
|
called {\it optimal}. Therefore, for solving LP problem the simplex
|
|
method tries to find its optimal basic solution.
|
|
|
|
Primal feasibility of some basic solution may be stated by simple
|
|
checking if all basic variables are within their bounds. Basic solution
|
|
is optimal if additionally the following optimality conditions are
|
|
satisfied for all non-basic variables:
|
|
\begin{center}
|
|
\begin{tabular}{lcc}
|
|
Status of $(x_N)_j$ & Minimization & Maximization \\
|
|
\hline
|
|
$(x_N)_j$ is free & $d_j = 0$ & $d_j = 0$ \\
|
|
$(x_N)_j$ is on its lower bound & $d_j \geq 0$ & $d_j \leq 0$ \\
|
|
$(x_N)_j$ is on its upper bound & $d_j \leq 0$ & $d_j \geq 0$ \\
|
|
\end{tabular}
|
|
\end{center}
|
|
In other words, basic solution is optimal if there is no non-basic
|
|
variable, which changing in the feasible direction (i.e. increasing if
|
|
it is free or on its lower bound, or decreasing if it is free or on its
|
|
upper bound) can improve (i.e. decrease in case of minimization or
|
|
increase in case of maximization) the objective function.
|
|
|
|
If all non-basic variables satisfy to the optimality conditions shown
|
|
above (independently on whether basic variables are within their bounds
|
|
or not), the basic solution is called {\it dual feasible}, otherwise it
|
|
is called {\it dual infeasible}.
|
|
|
|
It may happen that some LP problem has no primal feasible solution due
|
|
to incorrect\linebreak formulation --- this means that its constraints
|
|
conflict with each other. It also may happen that some LP problem has
|
|
unbounded solution again due to incorrect formulation --- this means
|
|
that some non-basic variable can improve the objective function, i.e.
|
|
the optimality conditions are violated, and at the same time this
|
|
variable can infinitely change in the feasible direction meeting
|
|
no resistance from basic variables. (May note that in the latter case
|
|
the LP problem has no dual feasible solution.)
|
|
|
|
\subsection{glp\_simplex --- solve LP problem with the primal or dual
|
|
simplex method}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_simplex(glp_prob *P, const glp_smcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_simplex| is a driver to the LP solver based on
|
|
the simplex method. This routine retrieves problem data from the
|
|
specified problem object, calls the solver to solve the problem
|
|
instance, and stores results of computations back into the problem
|
|
object.
|
|
|
|
The simplex solver has a set of control parameters. Values of the
|
|
control parameters can be passed in the structure \verb|glp_smcp|,
|
|
which the parameter \verb|parm| points to. For detailed description of
|
|
this structure see paragraph ``Control parameters'' below.
|
|
Before specifying some control parameters the application program
|
|
should initialize the structure \verb|glp_smcp| by default values of
|
|
all control parameters using the routine \verb|glp_init_smcp| (see the
|
|
next subsection). This is needed for backward compatibility, because in
|
|
the future there may appear new members in the structure
|
|
\verb|glp_smcp|.
|
|
|
|
The parameter \verb|parm| can be specified as \verb|NULL|, in which
|
|
case the solver uses default settings.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The LP problem instance has been successfully solved. (This code
|
|
does {\it not} necessarily mean that the solver has found optimal
|
|
solution. It only means that the solution process was successful.) \\
|
|
|
|
\verb|GLP_EBADB| & Unable to start the search, because the initial
|
|
basis specified in the problem object is invalid---the number of basic
|
|
(auxiliary and structural) variables is not the same as the number of
|
|
rows in the problem object.\\
|
|
|
|
\verb|GLP_ESING| & Unable to start the search, because the basis matrix
|
|
corresponding to the initial basis is singular within the working
|
|
precision.\\
|
|
|
|
\verb|GLP_ECOND| & Unable to start the search, because the basis matrix
|
|
corresponding to the initial basis is ill-conditioned, i.e. its
|
|
condition number is too large.\\
|
|
|
|
\verb|GLP_EBOUND| & Unable to start the search, because some
|
|
double-bounded (auxiliary or structural) variables have incorrect
|
|
bounds.\\
|
|
|
|
\verb|GLP_EFAIL| & The search was prematurely terminated due to the
|
|
solver failure.\\
|
|
|
|
\verb|GLP_EOBJLL| & The search was prematurely terminated, because the
|
|
objective function being maximized has reached its lower limit and
|
|
continues decreasing (the dual simplex only).\\
|
|
\end{retlist}
|
|
|
|
\begin{retlist}
|
|
\verb|GLP_EOBJUL| & The search was prematurely terminated, because the
|
|
objective function being minimized has reached its upper limit and
|
|
continues increasing (the dual simplex only).\\
|
|
|
|
\verb|GLP_EITLIM| & The search was prematurely terminated, because the
|
|
simplex iteration limit has been exceeded.\\
|
|
|
|
\verb|GLP_ETMLIM| & The search was prematurely terminated, because the
|
|
time limit has been exceeded.\\
|
|
|
|
\verb|GLP_ENOPFS| & The LP problem instance has no primal feasible
|
|
solution (only if the LP presolver is used).\\
|
|
|
|
\verb|GLP_ENODFS| & The LP problem instance has no dual feasible
|
|
solution (only if the LP presolver is used).\\
|
|
\end{retlist}
|
|
|
|
\para{Built-in LP presolver}
|
|
|
|
The simplex solver has {\it built-in LP presolver}. It is a subprogram
|
|
that transforms the original LP problem specified in the problem object
|
|
to an equivalent LP problem, which may be easier for solving with the
|
|
simplex method than the original one. This is attained mainly due to
|
|
reducing the problem size and improving its numeric properties (for
|
|
example, by removing some inactive constraints or by fixing some
|
|
non-basic variables). Once the transformed LP problem has been solved,
|
|
the presolver transforms its basic solution back to the corresponding
|
|
basic solution of the original problem.
|
|
|
|
Presolving is an optional feature of the routine \verb|glp_simplex|,
|
|
and by default it is disabled. In order to enable the LP presolver the
|
|
control parameter \verb|presolve| should be set to \verb|GLP_ON| (see
|
|
paragraph ``Control parameters'' below). Presolving may be used when
|
|
the problem instance is solved for the first time. However, on
|
|
performing re-optimization the presolver should be disabled.
|
|
|
|
The presolving procedure is transparent to the API user in the sense
|
|
that all necessary processing is performed internally, and a basic
|
|
solution of the original problem recovered by the presolver is the same
|
|
as if it were computed directly, i.e. without presolving.
|
|
|
|
Note that the presolver is able to recover only optimal solutions. If
|
|
a computed solution is infeasible or non-optimal, the corresponding
|
|
solution of the original problem cannot be recovered and therefore
|
|
remains undefined. If you need to know a basic solution even if it is
|
|
infeasible or non-optimal, the presolver should be disabled.
|
|
|
|
\para{Terminal output}
|
|
|
|
Solving large problem instances may take a long time, so the solver
|
|
reports some information about the current basic solution, which is
|
|
sent to the terminal. This information has the following format:
|
|
|
|
\begin{verbatim}
|
|
nnn: obj = xxx infeas = yyy (ddd)
|
|
\end{verbatim}
|
|
|
|
\noindent
|
|
where: `\verb|nnn|' is the iteration number, `\verb|xxx|' is the
|
|
current value of the objective function (it is unscaled and has correct
|
|
sign); `\verb|yyy|' is the current sum of primal or dual
|
|
infeasibilities (it is scaled and therefore may be used only for visual
|
|
estimating), `\verb|ddd|' is the current number of fixed basic
|
|
variables.
|
|
|
|
The symbol preceding the iteration number indicates which phase of the
|
|
simplex method is in effect:
|
|
|
|
{\it Blank} means that the solver is searching for primal feasible
|
|
solution using the primal simplex or for dual feasible solution using
|
|
the dual simplex;
|
|
|
|
{\it Asterisk} (\verb|*|) means that the solver is searching for
|
|
optimal solution using the primal simplex;
|
|
|
|
{\it Vertical dash} (\verb/|/) means that the solver is searching for
|
|
optimal solution using the dual simplex.
|
|
|
|
\para{Control parameters}
|
|
|
|
This paragraph describes all control parameters currently used in the
|
|
simplex solver. Symbolic names of control parameters are names of
|
|
corresponding members in the structure \verb|glp_smcp|.
|
|
|
|
\bigskip
|
|
|
|
{\tt int msg\_lev} (default: {\tt GLP\_MSG\_ALL})
|
|
|
|
Message level for terminal output:
|
|
|
|
\verb|GLP_MSG_OFF| --- no output;
|
|
|
|
\verb|GLP_MSG_ERR| --- error and warning messages only;
|
|
|
|
\verb|GLP_MSG_ON | --- normal output;
|
|
|
|
\verb|GLP_MSG_ALL| --- full output (including informational messages).
|
|
|
|
\bigskip
|
|
|
|
{\tt int meth} (default: {\tt GLP\_PRIMAL})
|
|
|
|
Simplex method option:
|
|
|
|
\verb|GLP_PRIMAL| --- use two-phase primal simplex;
|
|
|
|
\verb|GLP_DUAL | --- use two-phase dual simplex;
|
|
|
|
\verb|GLP_DUALP | --- use two-phase dual simplex, and if it fails,
|
|
switch to the primal simplex.
|
|
|
|
\bigskip
|
|
|
|
{\tt int pricing} (default: {\tt GLP\_PT\_PSE})
|
|
|
|
Pricing technique:
|
|
|
|
\verb|GLP_PT_STD| --- standard (``textbook'');
|
|
|
|
\verb|GLP_PT_PSE| --- projected steepest edge.
|
|
|
|
\bigskip
|
|
|
|
{\tt int r\_test} (default: {\tt GLP\_RT\_HAR})
|
|
|
|
Ratio test technique:
|
|
|
|
\verb|GLP_RT_STD| --- standard (``textbook'');
|
|
|
|
\verb|GLP_RT_HAR| --- Harris' two-pass ratio test.
|
|
|
|
\bigskip
|
|
|
|
{\tt double tol\_bnd} (default: {\tt 1e-7})
|
|
|
|
Tolerance used to check if the basic solution is primal feasible.
|
|
(Do not change this parameter without detailed understanding its
|
|
purpose.)
|
|
|
|
\newpage
|
|
|
|
{\tt double tol\_dj} (default: {\tt 1e-7})
|
|
|
|
Tolerance used to check if the basic solution is dual feasible.
|
|
(Do not change this parameter without detailed understanding its
|
|
purpose.)
|
|
|
|
\bigskip
|
|
|
|
{\tt double tol\_piv} (default: {\tt 1e-9})
|
|
|
|
Tolerance used to choose eligble pivotal elements of the simplex table.
|
|
(Do not change this parameter without detailed understanding its
|
|
purpose.)
|
|
|
|
\bigskip
|
|
|
|
{\tt double obj\_ll} (default: {\tt -DBL\_MAX})
|
|
|
|
Lower limit of the objective function. If the objective function
|
|
reaches this limit and continues decreasing, the solver terminates the
|
|
search. (Used in the dual simplex only.)
|
|
|
|
\bigskip
|
|
|
|
{\tt double obj\_ul} (default: {\tt +DBL\_MAX})
|
|
|
|
Upper limit of the objective function. If the objective function
|
|
reaches this limit and continues increasing, the solver terminates the
|
|
search. (Used in the dual simplex only.)
|
|
|
|
\bigskip
|
|
|
|
{\tt int it\_lim} (default: {\tt INT\_MAX})
|
|
|
|
Simplex iteration limit.
|
|
|
|
\bigskip
|
|
|
|
{\tt int tm\_lim} (default: {\tt INT\_MAX})
|
|
|
|
Searching time limit, in milliseconds.
|
|
|
|
\bigskip
|
|
|
|
{\tt int out\_frq} (default: {\tt 500})
|
|
|
|
Output frequency, in iterations. This parameter specifies how
|
|
frequently the solver sends information about the solution process to
|
|
the terminal.
|
|
|
|
\bigskip
|
|
|
|
{\tt int out\_dly} (default: {\tt 0})
|
|
|
|
Output delay, in milliseconds. This parameter specifies how long the
|
|
solver should delay sending information about the solution process to
|
|
the terminal.
|
|
|
|
\bigskip
|
|
|
|
{\tt int presolve} (default: {\tt GLP\_OFF})
|
|
|
|
LP presolver option:
|
|
|
|
\verb|GLP_ON | --- enable using the LP presolver;
|
|
|
|
\verb|GLP_OFF| --- disable using the LP presolver.
|
|
|
|
\newpage
|
|
|
|
\para{Example 1}
|
|
|
|
The following example main program reads LP problem instance in fixed
|
|
MPS format from file \verb|25fv47.mps|,\footnote{This instance in fixed
|
|
MPS format can be found in the Netlib LP collection; see
|
|
{\tt ftp://ftp.netlib.org/lp/data/}.} constructs an advanced initial
|
|
basis, solves the instance with the primal simplex method (by default),
|
|
and writes the solution to file \verb|25fv47.txt|.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
/* spxsamp1.c */
|
|
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
#include <glpk.h>
|
|
|
|
int main(void)
|
|
{ glp_prob *P;
|
|
P = glp_create_prob();
|
|
glp_read_mps(P, GLP_MPS_DECK, NULL, "25fv47.mps");
|
|
glp_adv_basis(P, 0);
|
|
glp_simplex(P, NULL);
|
|
glp_print_sol(P, "25fv47.txt");
|
|
glp_delete_prob(P);
|
|
return 0;
|
|
}
|
|
|
|
/* eof */
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
Below here is shown the terminal output from this example program.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
Reading problem data from `25fv47.mps'...
|
|
Problem: 25FV47
|
|
Objective: R0000
|
|
822 rows, 1571 columns, 11127 non-zeros
|
|
6919 records were read
|
|
Crashing...
|
|
Size of triangular part = 799
|
|
0: obj = 1.627307307e+04 infeas = 5.194e+04 (23)
|
|
200: obj = 1.474901610e+04 infeas = 1.233e+04 (19)
|
|
400: obj = 1.343909995e+04 infeas = 3.648e+03 (13)
|
|
600: obj = 1.756052217e+04 infeas = 4.179e+02 (7)
|
|
* 775: obj = 1.789251591e+04 infeas = 4.982e-14 (1)
|
|
* 800: obj = 1.663354510e+04 infeas = 2.857e-14 (1)
|
|
* 1000: obj = 1.024935068e+04 infeas = 1.958e-12 (1)
|
|
* 1200: obj = 7.860174791e+03 infeas = 2.810e-29 (1)
|
|
* 1400: obj = 6.642378184e+03 infeas = 2.036e-16 (1)
|
|
* 1600: obj = 6.037014568e+03 infeas = 0.000e+00 (1)
|
|
* 1800: obj = 5.662171307e+03 infeas = 6.447e-15 (1)
|
|
* 2000: obj = 5.528146165e+03 infeas = 9.764e-13 (1)
|
|
* 2125: obj = 5.501845888e+03 infeas = 0.000e+00 (1)
|
|
OPTIMAL SOLUTION FOUND
|
|
Writing basic solution to `25fv47.txt'...
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
\newpage
|
|
|
|
\para{Example 2}
|
|
|
|
The following example main program solves the same LP problem instance
|
|
as in Example 1 above, however, it uses the dual simplex method, which
|
|
starts from the standard initial basis.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
/* spxsamp2.c */
|
|
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
#include <glpk.h>
|
|
|
|
int main(void)
|
|
{ glp_prob *P;
|
|
glp_smcp parm;
|
|
P = glp_create_prob();
|
|
glp_read_mps(P, GLP_MPS_DECK, NULL, "25fv47.mps");
|
|
glp_init_smcp(&parm);
|
|
parm.meth = GLP_DUAL;
|
|
glp_simplex(P, &parm);
|
|
glp_print_sol(P, "25fv47.txt");
|
|
glp_delete_prob(P);
|
|
return 0;
|
|
}
|
|
|
|
/* eof */
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
Below here is shown the terminal output from this example program.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
Reading problem data from `25fv47.mps'...
|
|
Problem: 25FV47
|
|
Objective: R0000
|
|
822 rows, 1571 columns, 11127 non-zeros
|
|
6919 records were read
|
|
0: infeas = 1.223e+03 (516)
|
|
200: infeas = 7.000e+00 (471)
|
|
240: infeas = 1.106e-14 (461)
|
|
| 400: obj = -5.394267152e+03 infeas = 5.571e-16 (391)
|
|
| 600: obj = -4.586395752e+03 infeas = 1.389e-15 (340)
|
|
| 800: obj = -4.158268146e+03 infeas = 1.640e-15 (264)
|
|
| 1000: obj = -3.725320045e+03 infeas = 5.181e-15 (245)
|
|
| 1200: obj = -3.104802163e+03 infeas = 1.019e-14 (210)
|
|
| 1400: obj = -2.584190499e+03 infeas = 8.865e-15 (178)
|
|
| 1600: obj = -2.073852927e+03 infeas = 7.867e-15 (142)
|
|
| 1800: obj = -1.164037407e+03 infeas = 8.792e-15 (109)
|
|
| 2000: obj = -4.370590250e+02 infeas = 2.591e-14 (85)
|
|
| 2200: obj = 1.068240144e+03 infeas = 1.025e-13 (70)
|
|
| 2400: obj = 1.607481126e+03 infeas = 3.272e-14 (67)
|
|
| 2600: obj = 3.038230551e+03 infeas = 4.850e-14 (52)
|
|
| 2800: obj = 4.316238187e+03 infeas = 2.622e-14 (36)
|
|
| 3000: obj = 5.443842629e+03 infeas = 3.976e-15 (11)
|
|
| 3060: obj = 5.501845888e+03 infeas = 8.806e-15 (2)
|
|
OPTIMAL SOLUTION FOUND
|
|
Writing basic solution to `25fv47.txt'...
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_exact --- solve LP problem in exact arithmetic}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_exact(glp_prob *P, const glp_smcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_exact| is a tentative implementation of the
|
|
primal two-phase simplex method based on exact (rational) arithmetic.
|
|
It is similar to the routine \verb|glp_simplex|, however, for all
|
|
internal computations it uses arithmetic of rational numbers, which is
|
|
exact in mathematical sense, i.e. free of round-off errors unlike
|
|
floating-point arithmetic.
|
|
|
|
Note that the routine \verb|glp_exact| uses only two control parameters
|
|
passed in the structure \verb|glp_smcp|, namely, \verb|it_lim| and
|
|
\verb|tm_lim|.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The LP problem instance has been successfully solved. (This code
|
|
does {\it not} necessarily mean that the solver has found optimal
|
|
solution. It only means that the solution process was successful.) \\
|
|
|
|
\verb|GLP_EBADB| & Unable to start the search, because the initial basis
|
|
specified in the problem object is invalid---the number of basic
|
|
(auxiliary and structural) variables is not the same as the number of
|
|
rows in the problem object.\\
|
|
|
|
\verb|GLP_ESING| & Unable to start the search, because the basis matrix
|
|
corresponding to the initial basis is exactly singular.\\
|
|
|
|
\verb|GLP_EBOUND| & Unable to start the search, because some
|
|
double-bounded (auxiliary or structural) variables have incorrect
|
|
bounds.\\
|
|
|
|
\verb|GLP_EFAIL| & The problem instance has no rows/columns.\\
|
|
|
|
\verb|GLP_EITLIM| & The search was prematurely terminated, because the
|
|
simplex iteration limit has been exceeded.\\
|
|
|
|
\verb|GLP_ETMLIM| & The search was prematurely terminated, because the
|
|
time limit has been exceeded.\\
|
|
\end{retlist}
|
|
|
|
\para{Note}
|
|
|
|
Computations in exact arithmetic are very time-consuming, so solving
|
|
LP problem with the routine \verb|glp_exact| from the very beginning is
|
|
not a good idea. It is much better at first to find an optimal basis
|
|
with the routine \verb|glp_simplex| and only then to call
|
|
\verb|glp_exact|, in which case only a few simplex iterations need to
|
|
be performed in exact arithmetic.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_init\_smcp --- initialize simplex solver control
|
|
parameters}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_init_smcp(glp_smcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_init_smcp| initializes control parameters, which
|
|
are used by the simplex solver, with default values.
|
|
|
|
Default values of the control parameters are stored in
|
|
a \verb|glp_smcp| structure, which the parameter \verb|parm| points to.
|
|
|
|
\subsection{glp\_get\_status --- determine generic status of basic
|
|
solution}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_status(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_status| reports the generic status of the
|
|
current basic solution for the specified problem object as follows:
|
|
|
|
\verb|GLP_OPT | --- solution is optimal;
|
|
|
|
\verb|GLP_FEAS | --- solution is feasible;
|
|
|
|
\verb|GLP_INFEAS| --- solution is infeasible;
|
|
|
|
\verb|GLP_NOFEAS| --- problem has no feasible solution;
|
|
|
|
\verb|GLP_UNBND | --- problem has unbounded solution;
|
|
|
|
\verb|GLP_UNDEF | --- solution is undefined.
|
|
|
|
More detailed information about the status of basic solution can be
|
|
retrieved with the routines \verb|glp_get_prim_stat| and
|
|
\verb|glp_get_dual_stat|.
|
|
|
|
\subsection{glp\_get\_prim\_stat --- retrieve status of primal basic
|
|
solution}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_prim_stat(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_prim_stat| reports the status of the primal
|
|
basic solution for the specified problem object as follows:
|
|
|
|
\verb|GLP_UNDEF | --- primal solution is undefined;
|
|
|
|
\verb|GLP_FEAS | --- primal solution is feasible;
|
|
|
|
\verb|GLP_INFEAS| --- primal solution is infeasible;
|
|
|
|
\verb|GLP_NOFEAS| --- no primal feasible solution exists.
|
|
|
|
\subsection{glp\_get\_dual\_stat --- retrieve status of dual basic
|
|
solution}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_dual_stat(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_dual_stat| reports the status of the dual
|
|
basic solution for the specified problem object as follows:
|
|
|
|
\verb|GLP_UNDEF | --- dual solution is undefined;
|
|
|
|
\verb|GLP_FEAS | --- dual solution is feasible;
|
|
|
|
\verb|GLP_INFEAS| --- dual solution is infeasible;
|
|
|
|
\verb|GLP_NOFEAS| --- no dual feasible solution exists.
|
|
|
|
\subsection{glp\_get\_obj\_val --- retrieve objective value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_obj_val(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_obj_val| returns current value of the
|
|
objective function.
|
|
|
|
\subsection{glp\_get\_row\_stat --- retrieve row status}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_row_stat(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_stat| returns current status assigned to
|
|
the auxiliary variable associated with \verb|i|-th row as follows:
|
|
|
|
\verb|GLP_BS| --- basic variable;
|
|
|
|
\verb|GLP_NL| --- non-basic variable on its lower bound;
|
|
|
|
\verb|GLP_NU| --- non-basic variable on its upper bound;
|
|
|
|
\verb|GLP_NF| --- non-basic free (unbounded) variable;
|
|
|
|
\verb|GLP_NS| --- non-basic fixed variable.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_get\_row\_prim --- retrieve row primal value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_row_prim(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_prim| returns primal value of the
|
|
auxiliary variable associated with \verb|i|-th row.
|
|
|
|
\subsection{glp\_get\_row\_dual --- retrieve row dual value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_row_dual(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_dual| returns dual value (i.e. reduced
|
|
cost) of the auxiliary variable associated with \verb|i|-th row.
|
|
|
|
\subsection{glp\_get\_col\_stat --- retrieve column status}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_col_stat(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_stat| returns current status assigned to
|
|
the structural variable associated with \verb|j|-th column as follows:
|
|
|
|
\verb|GLP_BS| --- basic variable;
|
|
|
|
\verb|GLP_NL| --- non-basic variable on its lower bound;
|
|
|
|
\verb|GLP_NU| --- non-basic variable on its upper bound;
|
|
|
|
\verb|GLP_NF| --- non-basic free (unbounded) variable;
|
|
|
|
\verb|GLP_NS| --- non-basic fixed variable.
|
|
|
|
\subsection{glp\_get\_col\_prim --- retrieve column primal value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_col_prim(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_prim| returns primal value of the
|
|
structural variable associated with \verb|j|-th column.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_get\_col\_dual --- retrieve column dual value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_get_col_dual(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_dual| returns dual value (i.e. reduced
|
|
cost) of the structural variable associated with \verb|j|-th column.
|
|
|
|
\subsection{glp\_get\_unbnd\_ray --- determine variable causing
|
|
unboundedness}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_unbnd_ray(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_unbnd_ray| returns the number $k$ of
|
|
a variable, which causes primal or dual unboundedness.
|
|
If $1\leq k\leq m$, it is $k$-th auxiliary variable, and if
|
|
$m+1\leq k\leq m+n$, it is $(k-m)$-th structural variable, where $m$ is
|
|
the number of rows, $n$ is the number of columns in the problem object.
|
|
If such variable is not defined, the routine returns 0.
|
|
|
|
\para{Note}
|
|
|
|
If it is not exactly known which version of the simplex solver
|
|
detected unboundedness, i.e. whether the unboundedness is primal or
|
|
dual, it is sufficient to check the status of the variable
|
|
with the routine \verb|glp_get_row_stat| or \verb|glp_get_col_stat|.
|
|
If the variable is non-basic, the unboundedness is primal, otherwise,
|
|
if the variable is basic, the unboundedness is dual (the latter case
|
|
means that the problem has no primal feasible dolution).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Interior-point method routines}
|
|
|
|
{\it Interior-point methods} (also known as {\it barrier methods}) are
|
|
more modern and powerful numerical methods for large-scale linear
|
|
programming. Such methods are especially efficient for very sparse LP
|
|
problems and allow solving such problems much faster than the simplex
|
|
method.
|
|
|
|
In brief, the GLPK interior-point solver works as follows.
|
|
|
|
At first, the solver transforms the original LP to a {\it working} LP
|
|
in the standard format:
|
|
|
|
\medskip
|
|
|
|
\noindent
|
|
\hspace{.5in} minimize
|
|
$$z = c_1x_{m+1} + c_2x_{m+2} + \dots + c_nx_{m+n} + c_0 \eqno (2.4)$$
|
|
\hspace{.5in} subject to linear constraints
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}r@{\:}c@{\:}r@{\:}c@{\:}l}
|
|
a_{11}x_{m+1}&+&a_{12}x_{m+2}&+ \dots +&a_{1n}x_{m+n}&=&b_1 \\
|
|
a_{21}x_{m+1}&+&a_{22}x_{m+2}&+ \dots +&a_{2n}x_{m+n}&=&b_2 \\
|
|
\multicolumn{7}{c}
|
|
{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .} \\
|
|
a_{m1}x_{m+1}&+&a_{m2}x_{m+2}&+ \dots +&a_{mn}x_{m+n}&=&b_m \\
|
|
\end{array} \eqno (2.5)
|
|
$$
|
|
\hspace{.5in} and non-negative variables
|
|
$$x_1\geq 0,\ \ x_2\geq 0,\ \ \dots,\ \ x_n\geq 0 \eqno(2.6)$$
|
|
where: $z$ is the objective function; $x_1$, \dots, $x_n$ are variables;
|
|
$c_1$, \dots, $c_n$ are objective coefficients; $c_0$ is a constant term
|
|
of the objective function; $a_{11}$, \dots, $a_{mn}$ are constraint
|
|
coefficients; $b_1$, \dots, $b_m$ are right-hand sides.
|
|
|
|
Using vector and matrix notations the working LP (2.4)---(2.6) can be
|
|
written as follows:
|
|
$$z=c^Tx+c_0\ \rightarrow\ \min,\eqno(2.7)$$
|
|
$$Ax=b,\eqno(2.8)$$
|
|
$$x\geq 0,\eqno(2.9)$$
|
|
where: $x=(x_j)$ is $n$-vector of variables, $c=(c_j)$ is $n$-vector of
|
|
objective coefficients, $A=(a_{ij})$ is $m\times n$-matrix of
|
|
constraint coefficients, and $b=(b_i)$ is $m$-vector of right-hand
|
|
sides.
|
|
|
|
Karush--Kuhn--Tucker optimality conditions for LP (2.7)---(2.9) are the
|
|
following:
|
|
$$Ax=b,\eqno(2.10)$$
|
|
$$A^T\pi+\lambda=c,\eqno(2.11)$$
|
|
$$\lambda^Tx=0,\eqno(2.12)$$
|
|
$$x\geq 0,\ \ \lambda\geq 0,\eqno(2.13)$$
|
|
where:
|
|
$\pi$ is $m$-vector of Lagrange multipliers (dual variables) for
|
|
equality constraints (2.8),\linebreak $\lambda$ is $n$-vector of
|
|
Lagrange multipliers (dual variables) for non-negativity constraints
|
|
(2.9),\linebreak (2.10) is the primal feasibility condition, (2.11) is
|
|
the dual feasibility condition, (2.12) is the primal-dual
|
|
complementarity condition, and (2.13) is the non-negativity conditions.
|
|
|
|
The main idea of the primal-dual interior-point method is based on
|
|
finding a point in the primal-dual space (i.e. in the space of all
|
|
primal and dual variables $x$, $\pi$, and $\lambda$), which satisfies
|
|
to all optimality conditions (2.10)---(2.13). Obviously, $x$-component
|
|
of such point then provides an optimal solution to the working LP
|
|
(2.7)---(2.9).
|
|
|
|
To find the optimal point $(x^*,\pi^*,\lambda^*)$ the interior-point
|
|
method attempts to solve the system of equations (2.10)---(2.12), which
|
|
is closed in the sense that the number of variables $x_j$, $\pi_i$, and
|
|
$\lambda_j$ and the number equations are the same and equal to $m+2n$.
|
|
Due to condition (2.12) this system of equations is non-linear, so it
|
|
can be solved with a version of {\it Newton's method} provided with
|
|
additional rules to keep the current point within the positive orthant
|
|
as required by the non-negativity conditions (2.13).
|
|
|
|
Finally, once the optimal point $(x^*,\pi^*,\lambda^*)$ has been found,
|
|
the solver performs inverse transformations to recover corresponding
|
|
solution to the original LP passed to the solver from the application
|
|
program.
|
|
|
|
\subsection{glp\_interior --- solve LP problem with the interior-point
|
|
method}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_interior(glp_prob *P, const glp_iptcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_interior| is a driver to the LP solver based on
|
|
the primal-dual interior-point method. This routine retrieves problem
|
|
data from the specified problem object, calls the solver to solve the
|
|
problem instance, and stores results of computations back into the
|
|
problem object.
|
|
|
|
The interior-point solver has a set of control parameters. Values of
|
|
the control parameters can be passed in the structure \verb|glp_iptcp|,
|
|
which the parameter \verb|parm| points to. For detailed description of
|
|
this structure see paragraph ``Control parameters'' below. Before
|
|
specifying some control parameters the application program should
|
|
initialize the structure \verb|glp_iptcp| by default values of all
|
|
control parameters using the routine \verb|glp_init_iptcp| (see the
|
|
next subsection). This is needed for backward compatibility, because in
|
|
the future there may appear new members in the structure
|
|
\verb|glp_iptcp|.
|
|
|
|
The parameter \verb|parm| can be specified as \verb|NULL|, in which
|
|
case the solver uses default settings.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The LP problem instance has been successfully solved. (This code
|
|
does {\it not} necessarily mean that the solver has found optimal
|
|
solution. It only means that the solution process was successful.) \\
|
|
|
|
\verb|GLP_EFAIL| & The problem has no rows/columns.\\
|
|
|
|
\verb|GLP_ENOCVG| & Very slow convergence or divergence.\\
|
|
|
|
\verb|GLP_EITLIM| & Iteration limit exceeded.\\
|
|
|
|
\verb|GLP_EINSTAB| & Numerical instability on solving Newtonian
|
|
system.\\
|
|
\end{retlist}
|
|
|
|
\newpage
|
|
|
|
\para{Comments}
|
|
|
|
The routine \verb|glp_interior| implements an easy version of
|
|
the primal-dual interior-point method based on Mehrotra's
|
|
technique.\footnote{S. Mehrotra. On the implementation of a primal-dual
|
|
interior point method. SIAM J. on Optim., 2(4), pp. 575-601, 1992.}
|
|
|
|
Note that currently the GLPK interior-point solver does not include
|
|
many important features, in particular:
|
|
|
|
\vspace*{-8pt}
|
|
|
|
\begin{itemize}
|
|
\item it is not able to process dense columns. Thus, if the constraint
|
|
matrix of the LP problem has dense columns, the solving process may be
|
|
inefficient;
|
|
|
|
\item it has no features against numerical instability. For some LP
|
|
problems premature termination may happen if the matrix $ADA^T$ becomes
|
|
singular or ill-conditioned;
|
|
|
|
\item it is not able to identify the optimal basis, which corresponds
|
|
to the interior-point solution found.
|
|
\end{itemize}
|
|
|
|
\vspace*{-8pt}
|
|
|
|
\para{Terminal output}
|
|
|
|
Solving large LP problems may take a long time, so the solver reports
|
|
some information about every interior-point iteration,\footnote{Unlike
|
|
the simplex method the interior point method usually needs 30---50
|
|
iterations (independently on the problem size) in order to find an
|
|
optimal solution.} which is sent to the terminal. This information has
|
|
the following format:
|
|
|
|
\begin{verbatim}
|
|
nnn: obj = fff; rpi = ppp; rdi = ddd; gap = ggg
|
|
\end{verbatim}
|
|
|
|
\noindent where: \verb|nnn| is iteration number, \verb|fff| is the
|
|
current value of the objective function (in the case of maximization it
|
|
has wrong sign), \verb|ppp| is the current relative primal
|
|
infeasibility (cf. (2.10)):
|
|
$$\frac{\|Ax^{(k)}-b\|}{1+\|b\|},\eqno(2.14)$$
|
|
\verb|ddd| is the current relative dual infeasibility (cf. (2.11)):
|
|
$$\frac{\|A^T\pi^{(k)}+\lambda^{(k)}-c\|}{1+\|c\|},\eqno(2.15)$$
|
|
\verb|ggg| is the current primal-dual gap (cf. (2.12)):
|
|
$$\frac{|c^Tx^{(k)}-b^T\pi^{(k)}|}{1+|c^Tx^{(k)}|},\eqno(2.16)$$
|
|
and $[x^{(k)},\pi^{(k)},\lambda^{(k)}]$ is the current point on $k$-th
|
|
iteration, $k=0,1,2,\dots$\ . Note that all solution components are
|
|
internally scaled, so information sent to the terminal is suitable only
|
|
for visual inspection.
|
|
|
|
\newpage
|
|
|
|
\para{Control parameters}
|
|
|
|
This paragraph describes all control parameters currently used in the
|
|
interior-point solver. Symbolic names of control parameters are names of
|
|
corresponding members in the structure \verb|glp_iptcp|.
|
|
|
|
\bigskip
|
|
|
|
{\tt int msg\_lev} (default: {\tt GLP\_MSG\_ALL})
|
|
|
|
Message level for terminal output:
|
|
|
|
\verb|GLP_MSG_OFF|---no output;
|
|
|
|
\verb|GLP_MSG_ERR|---error and warning messages only;
|
|
|
|
\verb|GLP_MSG_ON |---normal output;
|
|
|
|
\verb|GLP_MSG_ALL|---full output (including informational messages).
|
|
|
|
\bigskip
|
|
|
|
{\tt int ord\_alg} (default: {\tt GLP\_ORD\_AMD})
|
|
|
|
Ordering algorithm used prior to Cholesky factorization:
|
|
|
|
\verb|GLP_ORD_NONE |---use natural (original) ordering;
|
|
|
|
\verb|GLP_ORD_QMD |---quotient minimum degree (QMD);
|
|
|
|
\verb|GLP_ORD_AMD |---approximate minimum degree (AMD);
|
|
|
|
\verb|GLP_ORD_SYMAMD|---approximate minimum degree (SYMAMD).
|
|
|
|
\bigskip
|
|
|
|
\para{Example}
|
|
|
|
The following main program reads LP problem instance in fixed MPS
|
|
format from file\linebreak \verb|25fv47.mps|,\footnote{This instance in
|
|
fixed MPS format can be found in the Netlib LP collection; see
|
|
{\tt ftp://ftp.netlib.org/lp/data/}.} solves it with the interior-point
|
|
solver, and writes the solution to file \verb|25fv47.txt|.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
/* iptsamp.c */
|
|
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
#include <glpk.h>
|
|
|
|
int main(void)
|
|
{ glp_prob *P;
|
|
P = glp_create_prob();
|
|
glp_read_mps(P, GLP_MPS_DECK, NULL, "25fv47.mps");
|
|
glp_interior(P, NULL);
|
|
glp_print_ipt(P, "25fv47.txt");
|
|
glp_delete_prob(P);
|
|
return 0;
|
|
}
|
|
|
|
/* eof */
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
\newpage
|
|
|
|
Below here is shown the terminal output from this example program.
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
Reading problem data from `25fv47.mps'...
|
|
Problem: 25FV47
|
|
Objective: R0000
|
|
822 rows, 1571 columns, 11127 non-zeros
|
|
6919 records were read
|
|
Original LP has 822 row(s), 1571 column(s), and 11127 non-zero(s)
|
|
Working LP has 821 row(s), 1876 column(s), and 10705 non-zero(s)
|
|
Matrix A has 10705 non-zeros
|
|
Matrix S = A*A' has 11895 non-zeros (upper triangle)
|
|
Minimal degree ordering...
|
|
Computing Cholesky factorization S = L'*L...
|
|
Matrix L has 35411 non-zeros
|
|
Guessing initial point...
|
|
Optimization begins...
|
|
0: obj = 1.823377629e+05; rpi = 1.3e+01; rdi = 1.4e+01; gap = 9.3e-01
|
|
1: obj = 9.260045192e+04; rpi = 5.3e+00; rdi = 5.6e+00; gap = 6.8e+00
|
|
2: obj = 3.596999742e+04; rpi = 1.5e+00; rdi = 1.2e+00; gap = 1.8e+01
|
|
3: obj = 1.989627568e+04; rpi = 4.7e-01; rdi = 3.0e-01; gap = 1.9e+01
|
|
4: obj = 1.430215557e+04; rpi = 1.1e-01; rdi = 8.6e-02; gap = 1.4e+01
|
|
5: obj = 1.155716505e+04; rpi = 2.3e-02; rdi = 2.4e-02; gap = 6.8e+00
|
|
6: obj = 9.660273208e+03; rpi = 6.7e-03; rdi = 4.6e-03; gap = 3.9e+00
|
|
7: obj = 8.694348283e+03; rpi = 3.7e-03; rdi = 1.7e-03; gap = 2.0e+00
|
|
8: obj = 8.019543639e+03; rpi = 2.4e-03; rdi = 3.9e-04; gap = 1.0e+00
|
|
9: obj = 7.122676293e+03; rpi = 1.2e-03; rdi = 1.5e-04; gap = 6.6e-01
|
|
10: obj = 6.514534518e+03; rpi = 6.1e-04; rdi = 4.3e-05; gap = 4.1e-01
|
|
11: obj = 6.361572203e+03; rpi = 4.8e-04; rdi = 2.2e-05; gap = 3.0e-01
|
|
12: obj = 6.203355508e+03; rpi = 3.2e-04; rdi = 1.7e-05; gap = 2.6e-01
|
|
13: obj = 6.032943411e+03; rpi = 2.0e-04; rdi = 9.3e-06; gap = 2.1e-01
|
|
14: obj = 5.796553021e+03; rpi = 9.8e-05; rdi = 3.2e-06; gap = 1.0e-01
|
|
15: obj = 5.667032431e+03; rpi = 4.4e-05; rdi = 1.1e-06; gap = 5.6e-02
|
|
16: obj = 5.613911867e+03; rpi = 2.5e-05; rdi = 4.1e-07; gap = 3.5e-02
|
|
17: obj = 5.560572626e+03; rpi = 9.9e-06; rdi = 2.3e-07; gap = 2.1e-02
|
|
18: obj = 5.537276001e+03; rpi = 5.5e-06; rdi = 8.4e-08; gap = 1.1e-02
|
|
19: obj = 5.522746942e+03; rpi = 2.2e-06; rdi = 4.0e-08; gap = 6.7e-03
|
|
20: obj = 5.509956679e+03; rpi = 7.5e-07; rdi = 1.8e-08; gap = 2.9e-03
|
|
21: obj = 5.504571733e+03; rpi = 1.6e-07; rdi = 5.8e-09; gap = 1.1e-03
|
|
22: obj = 5.502576367e+03; rpi = 3.4e-08; rdi = 1.0e-09; gap = 2.5e-04
|
|
23: obj = 5.502057119e+03; rpi = 8.1e-09; rdi = 3.0e-10; gap = 7.7e-05
|
|
24: obj = 5.501885996e+03; rpi = 9.4e-10; rdi = 1.2e-10; gap = 2.4e-05
|
|
25: obj = 5.501852464e+03; rpi = 1.4e-10; rdi = 1.2e-11; gap = 3.0e-06
|
|
26: obj = 5.501846549e+03; rpi = 1.4e-11; rdi = 1.2e-12; gap = 3.0e-07
|
|
27: obj = 5.501845954e+03; rpi = 1.4e-12; rdi = 1.2e-13; gap = 3.0e-08
|
|
28: obj = 5.501845895e+03; rpi = 1.5e-13; rdi = 1.2e-14; gap = 3.0e-09
|
|
OPTIMAL SOLUTION FOUND
|
|
Writing interior-point solution to `25fv47.txt'...
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_init\_iptcp --- initialize interior-point solver
|
|
control parameters}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_init_iptcp(glp_iptcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_init_iptcp| initializes control parameters, which
|
|
are used by the interior-point solver, with default values.
|
|
|
|
Default values of the control parameters are stored in the structure
|
|
\verb|glp_iptcp|, which the parameter \verb|parm| points to.
|
|
|
|
\subsection{glp\_ipt\_status --- determine solution status}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_ipt_status(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_status| reports the status of a solution
|
|
found by the interior-point solver as follows:
|
|
|
|
\verb|GLP_UNDEF | --- interior-point solution is undefined;
|
|
|
|
\verb|GLP_OPT | --- interior-point solution is optimal;
|
|
|
|
\verb|GLP_INFEAS| --- interior-point solution is infeasible;
|
|
|
|
\verb|GLP_NOFEAS| --- no feasible primal-dual solution exists.
|
|
|
|
\subsection{glp\_ipt\_obj\_val --- retrieve objective value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_ipt_obj_val(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_obj_val| returns value of the objective
|
|
function for interior-point solution.
|
|
|
|
\subsection{glp\_ipt\_row\_prim --- retrieve row primal value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_ipt_row_prim(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_row_prim| returns primal value of the
|
|
auxiliary variable associated with \verb|i|-th row.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_ipt\_row\_dual --- retrieve row dual value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_ipt_row_dual(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_row_dual| returns dual value (i.e. reduced
|
|
cost) of the auxiliary variable associated with \verb|i|-th row.
|
|
|
|
\subsection{glp\_ipt\_col\_prim --- retrieve column primal value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_ipt_col_prim(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_col_prim| returns primal value of the
|
|
structural variable associated with \verb|j|-th column.
|
|
|
|
\subsection{glp\_ipt\_col\_dual --- retrieve column dual value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_ipt_col_dual(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_ipt_col_dual| returns dual value (i.e. reduced
|
|
cost) of the structural variable associated with \verb|j|-th column.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Mixed integer programming routines}
|
|
|
|
\subsection{glp\_set\_col\_kind --- set (change) column kind}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_col_kind(glp_prob *P, int j, int kind);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_col_kind| sets (changes) the kind of
|
|
\verb|j|-th column (structural variable) as specified by the parameter
|
|
\verb|kind|:
|
|
|
|
\verb|GLP_CV| --- continuous variable;
|
|
|
|
\verb|GLP_IV| --- integer variable;
|
|
|
|
\verb|GLP_BV| --- binary variable.
|
|
|
|
Setting a column to \verb|GLP_BV| has the same effect as if it were
|
|
set to \verb|GLP_IV|, its lower bound were set 0, and its upper bound
|
|
were set to 1.
|
|
|
|
\subsection{glp\_get\_col\_kind --- retrieve column kind}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_col_kind(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_kind| returns the kind of \verb|j|-th
|
|
column (structural variable) as follows:
|
|
|
|
\verb|GLP_CV| --- continuous variable;
|
|
|
|
\verb|GLP_IV| --- integer variable;
|
|
|
|
\verb|GLP_BV| --- binary variable.
|
|
|
|
\subsection{glp\_get\_num\_int --- retrieve number of integer columns}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_num_int(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_num_int| returns the number of columns
|
|
(structural variables), which are marked as integer. Note that this
|
|
number {\it does} include binary columns.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_get\_num\_bin --- retrieve number of binary columns}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_num_bin(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_num_bin| returns the number of columns
|
|
(structural variables), which are marked as integer and whose lower
|
|
bound is zero and upper bound is one.
|
|
|
|
\subsection{glp\_intopt --- solve MIP problem with the branch-and-cut
|
|
method}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_intopt(glp_prob *P, const glp_iocp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_intopt| is a driver to the MIP solver based on
|
|
the branch-and-cut method, which is a hybrid of branch-and-bound and
|
|
cutting plane methods.
|
|
|
|
If the presolver is disabled (see paragraph ``Control parameters''
|
|
below), on entry to the routine \verb|glp_intopt| the problem object,
|
|
which the parameter \verb|mip| points to, should contain optimal
|
|
solution to LP relaxation (it can be obtained, for example, with the
|
|
routine \verb|glp_simplex|). Otherwise, if the presolver is enabled, it
|
|
is not necessary.
|
|
|
|
The MIP solver has a set of control parameters. Values of the control
|
|
parameters can be passed in the structure \verb|glp_iocp|, which the
|
|
parameter \verb|parm| points to. For detailed description of this
|
|
structure see paragraph ``Control parameters'' below. Before specifying
|
|
some control parameters the application program should initialize the
|
|
structure \verb|glp_iocp| by default values of all control parameters
|
|
using the routine \verb|glp_init_iocp| (see the next subsection). This
|
|
is needed for backward compatibility, because in the future there may
|
|
appear new members in the structure \verb|glp_iocp|.
|
|
|
|
The parameter \verb|parm| can be specified as \verb|NULL|, in which case
|
|
the solver uses default settings.
|
|
|
|
Note that the GLPK branch-and-cut solver is not perfect, so it is
|
|
unable to solve hard or very large scale MIP instances for a reasonable
|
|
time.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The MIP problem instance has been successfully solved. (This code
|
|
does {\it not} necessarily mean that the solver has found optimal
|
|
solution. It only means that the solution process was successful.) \\
|
|
|
|
\verb|GLP_EBOUND| & Unable to start the search, because some
|
|
double-bounded variables have incorrect bounds or some integer
|
|
variables have non-integer (fractional) bounds.\\
|
|
|
|
\verb|GLP_EROOT| & Unable to start the search, because optimal basis
|
|
for initial LP relaxation is not provided. (This code may appear only
|
|
if the presolver is disabled.)\\
|
|
|
|
\verb|GLP_ENOPFS| & Unable to start the search, because LP relaxation
|
|
of the MIP problem instance has no primal feasible solution. (This code
|
|
may appear only if the presolver is enabled.)\\
|
|
\end{retlist}
|
|
|
|
\newpage
|
|
|
|
\begin{retlist}
|
|
\verb|GLP_ENODFS| & Unable to start the search, because LP relaxation
|
|
of the MIP problem instance has no dual feasible solution. In other
|
|
word, this code means that if the LP relaxation has at least one primal
|
|
feasible solution, its optimal solution is unbounded, so if the MIP
|
|
problem has at least one integer feasible solution, its (integer)
|
|
optimal solution is also unbounded. (This code may appear only if the
|
|
presolver is enabled.)\\
|
|
|
|
\verb|GLP_EFAIL| & The search was prematurely terminated due to the
|
|
solver failure.\\
|
|
|
|
\verb|GLP_EMIPGAP| & The search was prematurely terminated, because the
|
|
relative mip gap tolerance has been reached.\\
|
|
|
|
\verb|GLP_ETMLIM| & The search was prematurely terminated, because the
|
|
time limit has been exceeded.\\
|
|
|
|
\verb|GLP_ESTOP| & The search was prematurely terminated by application.
|
|
(This code may appear only if the advanced solver interface is used.)\\
|
|
\end{retlist}
|
|
|
|
\para{Built-in MIP presolver}
|
|
|
|
The branch-and-cut solver has {\it built-in MIP presolver}. It is
|
|
a subprogram that transforms the original MIP problem specified in the
|
|
problem object to an equivalent MIP problem, which may be easier for
|
|
solving with the branch-and-cut method than the original one. For
|
|
example, the presolver can remove redundant constraints and variables,
|
|
whose optimal values are known, perform bound and coefficient reduction,
|
|
etc. Once the transformed MIP problem has been solved, the presolver
|
|
transforms its solution back to corresponding solution of the original
|
|
problem.
|
|
|
|
Presolving is an optional feature of the routine \verb|glp_intopt|, and
|
|
by default it is disabled. In order to enable the MIP presolver, the
|
|
control parameter \verb|presolve| should be set to \verb|GLP_ON| (see
|
|
paragraph ``Control parameters'' below).
|
|
|
|
\para{Advanced solver interface}
|
|
|
|
The routine \verb|glp_intopt| allows the user to control the
|
|
branch-and-cut search by passing to the solver a user-defined callback
|
|
routine. For more details see Chapter ``Branch-and-Cut API Routines''.
|
|
|
|
\para{Terminal output}
|
|
|
|
Solving a MIP problem may take a long time, so the solver reports some
|
|
information about best known solutions, which is sent to the terminal.
|
|
This information has the following format:
|
|
|
|
\begin{verbatim}
|
|
+nnn: mip = xxx <rho> yyy gap (ppp; qqq)
|
|
\end{verbatim}
|
|
|
|
\noindent
|
|
where: `\verb|nnn|' is the simplex iteration number; `\verb|xxx|' is a
|
|
value of the objective function for the best known integer feasible
|
|
solution (if no integer feasible solution has been found yet,
|
|
`\verb|xxx|' is the text `\verb|not found yet|'); `\verb|rho|' is the
|
|
string `\verb|>=|' (in case of minimization) or `\verb|<=|' (in case of
|
|
maximization); `\verb|yyy|' is a global bound for exact integer optimum
|
|
(i.e. the exact integer optimum is always in the range from `\verb|xxx|'
|
|
to `\verb|yyy|'); `\verb|gap|' is the relative mip gap, in percents,
|
|
computed as $gap=|xxx-yyy|/(|xxx|+{\tt DBL\_EPSILON})\cdot 100\%$ (if
|
|
$gap$ is greater than $999.9\%$, it is not printed); `\verb|ppp|' is the
|
|
number of subproblems in the active list, `\verb|qqq|' is the number of
|
|
subproblems which have been already fathomed and therefore removed from
|
|
the branch-and-bound search tree.
|
|
|
|
\newpage
|
|
|
|
\subsubsection{Control parameters}
|
|
|
|
This paragraph describes all control parameters currently used in the
|
|
MIP solver. Symbolic names of control parameters are names of
|
|
corresponding members in the structure \verb|glp_iocp|.
|
|
|
|
\bigskip\vspace*{-2pt}
|
|
|
|
{\tt int msg\_lev} (default: {\tt GLP\_MSG\_ALL})
|
|
|
|
Message level for terminal output:
|
|
|
|
\verb|GLP_MSG_OFF| --- no output;
|
|
|
|
\verb|GLP_MSG_ERR| --- error and warning messages only;
|
|
|
|
\verb|GLP_MSG_ON | --- normal output;
|
|
|
|
\verb|GLP_MSG_ALL| --- full output (including informational messages).
|
|
|
|
\bigskip\vspace*{-2pt}
|
|
|
|
{\tt int br\_tech} (default: {\tt GLP\_BR\_DTH})
|
|
|
|
Branching technique option:
|
|
|
|
\verb|GLP_BR_FFV| --- first fractional variable;
|
|
|
|
\verb|GLP_BR_LFV| --- last fractional variable;
|
|
|
|
\verb|GLP_BR_MFV| --- most fractional variable;
|
|
|
|
\verb|GLP_BR_DTH| --- heuristic by Driebeck and Tomlin;
|
|
|
|
\verb|GLP_BR_PCH| --- hybrid pseudo-cost heuristic.
|
|
|
|
\bigskip\vspace*{-2pt}
|
|
|
|
{\tt int bt\_tech} (default: {\tt GLP\_BT\_BLB})
|
|
|
|
Backtracking technique option:
|
|
|
|
\verb|GLP_BT_DFS| --- depth first search;
|
|
|
|
\verb|GLP_BT_BFS| --- breadth first search;
|
|
|
|
\verb|GLP_BT_BLB| --- best local bound;
|
|
|
|
\verb|GLP_BT_BPH| --- best projection heuristic.
|
|
|
|
\bigskip\vspace*{-2pt}
|
|
|
|
{\tt int pp\_tech} (default: {\tt GLP\_PP\_ALL})
|
|
|
|
Preprocessing technique option:
|
|
|
|
\verb|GLP_PP_NONE| --- disable preprocessing;
|
|
|
|
\verb|GLP_PP_ROOT| --- perform preprocessing only on the root level;
|
|
|
|
\verb|GLP_PP_ALL | --- perform preprocessing on all levels.
|
|
|
|
\bigskip\vspace*{-2pt}
|
|
|
|
{\tt int sr\_heur} (default: {\tt GLP\_ON})
|
|
|
|
Simple rounding heuristic option:
|
|
|
|
\verb|GLP_ON | --- enable applying the simple rounding heuristic;
|
|
|
|
\verb|GLP_OFF| --- disable applying the simple rounding heuristic.
|
|
|
|
\newpage
|
|
|
|
{\tt int fp\_heur} (default: {\tt GLP\_OFF})
|
|
|
|
Feasibility pump heuristic option:
|
|
|
|
\verb|GLP_ON | --- enable applying the feasibility pump heuristic;
|
|
|
|
\verb|GLP_OFF| --- disable applying the feasibility pump heuristic.
|
|
|
|
\bigskip
|
|
|
|
{\tt int ps\_heur} (default: {\tt GLP\_OFF})
|
|
|
|
Proximity search heuristic\footnote{The Fischetti--Monaci Proximity
|
|
Search (a.k.a. Proxy) heuristic. This algorithm is often capable of
|
|
rapidly improving a feasible solution of a MIP problem with binary
|
|
variables. It allows to quickly obtain suboptimal solutions in some
|
|
problems which take too long time to be solved to optimality.} option:
|
|
|
|
\verb|GLP_ON | --- enable applying the proximity search heuristic;
|
|
|
|
\verb|GLP_OFF| --- disable applying the proximity search pump heuristic.
|
|
|
|
\bigskip
|
|
|
|
{\tt int ps\_tm\_lim} (default: {\tt 60000})
|
|
|
|
Time limit, in milliseconds, for the proximity search heuristic (see
|
|
above).
|
|
|
|
\bigskip
|
|
|
|
{\tt int gmi\_cuts} (default: {\tt GLP\_OFF})
|
|
|
|
Gomory's mixed integer cut option:
|
|
|
|
\verb|GLP_ON | --- enable generating Gomory's cuts;
|
|
|
|
\verb|GLP_OFF| --- disable generating Gomory's cuts.
|
|
|
|
\bigskip
|
|
|
|
{\tt int mir\_cuts} (default: {\tt GLP\_OFF})
|
|
|
|
Mixed integer rounding (MIR) cut option:
|
|
|
|
\verb|GLP_ON | --- enable generating MIR cuts;
|
|
|
|
\verb|GLP_OFF| --- disable generating MIR cuts.
|
|
|
|
\bigskip
|
|
|
|
{\tt int cov\_cuts} (default: {\tt GLP\_OFF})
|
|
|
|
Mixed cover cut option:
|
|
|
|
\verb|GLP_ON | --- enable generating mixed cover cuts;
|
|
|
|
\verb|GLP_OFF| --- disable generating mixed cover cuts.
|
|
|
|
\bigskip
|
|
|
|
{\tt int clq\_cuts} (default: {\tt GLP\_OFF})
|
|
|
|
Clique cut option:
|
|
|
|
\verb|GLP_ON | --- enable generating clique cuts;
|
|
|
|
\verb|GLP_OFF| --- disable generating clique cuts.
|
|
|
|
\newpage
|
|
|
|
{\tt double tol\_int} (default: {\tt 1e-5})
|
|
|
|
Absolute tolerance used to check if optimal solution to the current LP
|
|
relaxation is integer feasible. (Do not change this parameter without
|
|
detailed understanding its purpose.)
|
|
|
|
\bigskip
|
|
|
|
{\tt double tol\_obj} (default: {\tt 1e-7})
|
|
|
|
Relative tolerance used to check if the objective value in optimal
|
|
solution to the current LP relaxation is not better than in the best
|
|
known integer feasible solution. (Do not change this parameter without
|
|
detailed understanding its purpose.)
|
|
|
|
\bigskip
|
|
|
|
{\tt double mip\_gap} (default: {\tt 0.0})
|
|
|
|
The relative mip gap tolerance. If the relative mip gap for currently
|
|
known best integer feasible solution falls below this tolerance, the
|
|
solver terminates the search. This allows obtainig suboptimal integer
|
|
feasible solutions if solving the problem to optimality takes too long
|
|
time.
|
|
|
|
\bigskip
|
|
|
|
{\tt int tm\_lim} (default: {\tt INT\_MAX})
|
|
|
|
Searching time limit, in milliseconds.
|
|
|
|
\bigskip
|
|
|
|
{\tt int out\_frq} (default: {\tt 5000})
|
|
|
|
Output frequency, in milliseconds. This parameter specifies how
|
|
frequently the solver sends information about the solution process to
|
|
the terminal.
|
|
|
|
\bigskip
|
|
|
|
{\tt int out\_dly} (default: {\tt 10000})
|
|
|
|
Output delay, in milliseconds. This parameter specifies how long the
|
|
solver should delay sending information about solution of the current
|
|
LP relaxation with the simplex method to the terminal.
|
|
|
|
\bigskip
|
|
|
|
{\tt void (*cb\_func)(glp\_tree *tree, void *info)}
|
|
(default: {\tt NULL})
|
|
|
|
Entry point to the user-defined callback routine. \verb|NULL| means
|
|
the advanced solver interface is not used. For more details see Chapter
|
|
``Branch-and-Cut API Routines''.
|
|
|
|
\bigskip
|
|
|
|
{\tt void *cb\_info} (default: {\tt NULL})
|
|
|
|
Transit pointer passed to the routine \verb|cb_func| (see above).
|
|
|
|
\bigskip
|
|
|
|
{\tt int cb\_size} (default: {\tt 0})
|
|
|
|
The number of extra (up to 256) bytes allocated for each node of the
|
|
branch-and-bound tree to store application-specific data. On creating
|
|
a node these bytes are initialized by binary zeros.
|
|
|
|
\bigskip
|
|
|
|
{\tt int presolve} (default: {\tt GLP\_OFF})
|
|
|
|
MIP presolver option:
|
|
|
|
\verb|GLP_ON | --- enable using the MIP presolver;
|
|
|
|
\verb|GLP_OFF| --- disable using the MIP presolver.
|
|
|
|
\newpage
|
|
|
|
{\tt int binarize} (default: {\tt GLP\_OFF})
|
|
|
|
Binarization option (used only if the presolver is enabled):
|
|
|
|
\verb|GLP_ON | --- replace general integer variables by binary ones;
|
|
|
|
\verb|GLP_OFF| --- do not use binarization.
|
|
|
|
\subsection{glp\_init\_iocp --- initialize integer optimizer control
|
|
parameters}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_init_iocp(glp_iocp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_init_iocp| initializes control parameters, which
|
|
are used by the branch-and-cut solver, with default values.
|
|
|
|
Default values of the control parameters are stored in
|
|
a \verb|glp_iocp| structure, which the parameter \verb|parm| points to.
|
|
|
|
\subsection{glp\_mip\_status --- determine status of MIP solution}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_mip_status(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_mip_status| reports the status of a MIP solution
|
|
found by the MIP solver as follows:
|
|
|
|
\verb|GLP_UNDEF | --- MIP solution is undefined;
|
|
|
|
\verb|GLP_OPT | --- MIP solution is integer optimal;
|
|
|
|
\verb|GLP_FEAS | --- MIP solution is integer feasible, however, its
|
|
optimality (or non-optimality) has not been proven, perhaps due to
|
|
premature termination of the search;
|
|
|
|
\verb|GLP_NOFEAS| --- problem has no integer feasible solution (proven
|
|
by the solver).
|
|
|
|
\subsection{glp\_mip\_obj\_val --- retrieve objective value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_mip_obj_val(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_mip_obj_val| returns value of the objective
|
|
function for MIP solution.
|
|
|
|
\subsection{glp\_mip\_row\_val --- retrieve row value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_mip_row_val(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_mip_row_val| returns value of the auxiliary
|
|
variable associated with \verb|i|-th row for MIP solution.
|
|
|
|
\subsection{glp\_mip\_col\_val --- retrieve column value}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
double glp_mip_col_val(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_mip_col_val| returns value of the structural
|
|
variable associated with \verb|j|-th column for MIP solution.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Additional routines}
|
|
|
|
\subsection{glp\_check\_kkt --- check feasibility/optimality
|
|
conditions}
|
|
|
|
\synopsis
|
|
|
|
{\parskip=0pt
|
|
\tt void glp\_check\_kkt(glp\_prob *P, int sol, int cond,
|
|
double *ae\_max, int *ae\_ind,
|
|
|
|
\hspace{105pt}double *re\_max, int *re\_ind);}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_check_kkt| allows to check
|
|
feasibility/optimality conditions for the current solution stored in
|
|
the specified problem object. (For basic and interior-point solutions
|
|
these conditions are known as {\it Karush--Kuhn--Tucker optimality
|
|
conditions}.)
|
|
|
|
The parameter \verb|sol| specifies which solution should be checked:
|
|
|
|
\verb|GLP_SOL| --- basic solution;
|
|
|
|
\verb|GLP_IPT| --- interior-point solution;
|
|
|
|
\verb|GLP_MIP| --- mixed integer solution.
|
|
|
|
The parameter \verb|cond| specifies which condition should be checked:
|
|
|
|
\verb|GLP_KKT_PE| --- check primal equality constraints (KKT.PE);
|
|
|
|
\verb|GLP_KKT_PB| --- check primal bound constraints (KKT.PB);
|
|
|
|
\verb|GLP_KKT_DE| --- check dual equality constraints (KKT.DE). This
|
|
conditions can be checked only for basic or interior-point solution;
|
|
|
|
\verb|GLP_KKT_DB| --- check dual bound constraints (KKT.DB). This
|
|
conditions can be checked only for basic or interior-point solution.
|
|
|
|
Detailed explanations of these conditions are given below in paragraph
|
|
``Background''.
|
|
|
|
On exit the routine stores the following information to locations
|
|
specified by parameters \verb|ae_max|, \verb|ae_ind|, \verb|re_max|,
|
|
and \verb|re_ind| (if some parameter is a null pointer, corresponding
|
|
information is not stored):
|
|
|
|
\verb|ae_max| --- largest absolute error;
|
|
|
|
\verb|ae_ind| --- number of row (KKT.PE), column (KKT.DE), or variable
|
|
(KKT.PB, KKT.DB) with the largest absolute error;
|
|
|
|
\verb|re_max| --- largest relative error;
|
|
|
|
\verb|re_ind| --- number of row (KKT.PE), column (KKT.DE), or variable
|
|
(KKT.PB, KKT.DB) with the largest relative error.
|
|
|
|
Row (auxiliary variable) numbers are in the range 1 to $m$, where $m$
|
|
is the number of rows in the problem object. Column (structural
|
|
variable) numbers are in the range 1 to $n$, where $n$ is the number
|
|
of columns in the problem object. Variable numbers are in the range
|
|
1 to $m+n$, where variables with numbers 1 to $m$ correspond to rows,
|
|
and variables with numbers $m+1$ to $m+n$ correspond to columns. If
|
|
the error reported is exact zero, corresponding row, column or variable
|
|
number is set to zero.
|
|
|
|
\para{Background}
|
|
|
|
\def\arraystretch{1.5}
|
|
|
|
The first condition checked by the routine is the following:
|
|
$$x_R - A x_S = 0, \eqno{\rm (KKT.PE)}$$
|
|
where $x_R$ is the subvector of auxiliary variables (rows), $x_S$ is
|
|
the subvector of structural variables (columns), $A$ is the constraint
|
|
matrix. This condition expresses the requirement that all primal
|
|
variables should satisfy to the system of equality constraints of the
|
|
original LP problem. In case of exact arithmetic this condition would
|
|
be satisfied for any basic solution; however, in case of inexact
|
|
(floating-point) arithmetic, this condition shows how accurate the
|
|
primal solution is, that depends on accuracy of a representation of the
|
|
basis matrix used by the simplex method, or on accuracy provided by the
|
|
interior-point method.
|
|
|
|
To check the condition (KKT.PE) the routine computes the vector of
|
|
residuals:
|
|
$$g = x_R - A x_S,$$
|
|
and determines component of this vector that correspond to largest
|
|
absolute and relative errors:
|
|
$${\tt ae\_max}=\max_{1\leq i\leq m}|g_i|,$$
|
|
$${\tt re\_max}=\max_{1\leq i\leq m}\frac{|g_i|}{1+|(x_R)_i|}.$$
|
|
|
|
The second condition checked by the routine is the following:
|
|
$$l_k \leq x_k \leq u_k {\rm \ \ \ for\ all}\ k=1,\dots,m+n,
|
|
\eqno{\rm (KKT.PB)}$$
|
|
where $x_k$ is auxiliary ($1\leq k\leq m$) or structural
|
|
($m+1\leq k\leq m+n$) variable, $l_k$ and $u_k$ are, respectively,
|
|
lower and upper bounds of the variable $x_k$ (including cases of
|
|
infinite bounds). This condition expresses the requirement that all
|
|
primal variables shoudl satisfy to bound constraints of the original
|
|
LP problem. In case of basic solution all non-basic variables are
|
|
placed on their active bounds, so actually the condition (KKT.PB) needs
|
|
to be checked for basic variables only. If the primal solution has
|
|
sufficient accuracy, this condition shows its primal feasibility.
|
|
|
|
To check the condition (KKT.PB) the routine computes a vector of
|
|
residuals:
|
|
$$
|
|
h_k = \left\{
|
|
\begin{array}{ll}
|
|
0, & {\rm if}\ l_k \leq x_k \leq u_k \\
|
|
x_k - l_k, & {\rm if}\ x_k < l_k \\
|
|
x_k - u_k, & {\rm if}\ x_k > u_k \\
|
|
\end{array}
|
|
\right.
|
|
$$
|
|
for all $k=1,\dots,m+n$, and determines components of this vector that
|
|
correspond to largest absolute and relative errors:
|
|
$${\tt ae\_max}=\max_{1\leq k \leq m+n}|h_k|,$$
|
|
$${\tt re\_max}=\max_{1\leq k \leq m+n}\frac{|h_k|}{1+|x_k|}.$$
|
|
|
|
The third condition checked by the routine is:
|
|
$${\rm grad}\;Z = c = (\tilde{A})^T \pi + d,$$
|
|
where $Z$ is the objective function, $c$ is the vector of objective
|
|
coefficients, $(\tilde{A})^T$ is a matrix transposed to the expanded
|
|
constraint matrix $\tilde{A} = (I|-A)$, $\pi$ is a vector of Lagrange
|
|
multipliers that correspond to equality constraints of the original LP
|
|
problem, $d$ is a vector of Lagrange multipliers that correspond to
|
|
bound constraints for all (auxiliary and structural) variables of the
|
|
original LP problem. Geometrically the third condition expresses the
|
|
requirement that the gradient of the objective function should belong
|
|
to the orthogonal complement of a linear subspace defined by the
|
|
equality and active bound constraints, i.e. that the gradient is
|
|
a linear combination of normals to the constraint hyperplanes, where
|
|
Lagrange multipliers $\pi$ and $d$ are coefficients of that linear
|
|
combination.
|
|
|
|
To eliminate the vector $\pi$ rewrite the third condition as:
|
|
$$
|
|
\left(\begin{array}{@{}c@{}}I \\ -A^T\end{array}\right) \pi =
|
|
\left(\begin{array}{@{}c@{}}d_R \\ d_S\end{array}\right) +
|
|
\left(\begin{array}{@{}c@{}}c_R \\ c_S\end{array}\right),
|
|
$$
|
|
or, equivalently,
|
|
$$
|
|
\left\{
|
|
\begin{array}{r@{}c@{}c}
|
|
\pi + d_R&\ =\ &c_R, \\
|
|
-A^T\pi + d_S&\ =\ &c_S. \\
|
|
\end{array}
|
|
\right.
|
|
$$
|
|
|
|
Then substituting the vector $\pi$ from the first equation into the
|
|
second we finally have:
|
|
$$A^T (d_R - c_R) + (d_S - c_S) = 0, \eqno{\rm(KKT.DE)}$$
|
|
where $d_R$ is the subvector of reduced costs of auxiliary variables
|
|
(rows), $d_S$ is the subvector of reduced costs of structural variables
|
|
(columns), $c_R$ and $c_S$ are subvectors of objective coefficients at,
|
|
respectively, auxiliary and structural variables, $A^T$ is a matrix
|
|
transposed to the constraint matrix of the original LP problem. In case
|
|
of exact arithmetic this condition would be satisfied for any basic
|
|
solution; however, in case of inexact (floating-point) arithmetic, this
|
|
condition shows how accurate the dual solution is, that depends on
|
|
accuracy of a representation of the basis matrix used by the simplex
|
|
method, or on accuracy provided by the interior-point method.
|
|
|
|
To check the condition (KKT.DE) the routine computes a vector of
|
|
residuals:
|
|
$$u = A^T (d_R - c_R) + (d_S - c_S),$$
|
|
and determines components of this vector that correspond to largest
|
|
absolute and relative errors:
|
|
$${\tt ae\_max}=\max_{1\leq j\leq n}|u_j|,$$
|
|
$${\tt re\_max}=\max_{1\leq j\leq n}\frac{|u_j|}{1+|(d_S)_j-(c_S)_j|}.$$
|
|
|
|
\newpage
|
|
|
|
The fourth condition checked by the routine is the following:
|
|
$$
|
|
\left\{
|
|
\begin{array}{l@{\ }r@{\ }c@{\ }c@{\ }c@{\ }l@{\ }c@{\ }c@{\ }c@{\ }l}
|
|
{\rm if} & -\infty & < & x_k & < & +\infty,
|
|
& {\rm then} & d_k & = & 0 \\
|
|
{\rm if} & l_k & \leq & x_k & < & +\infty,
|
|
& {\rm then} & d_k & \geq & 0\ {\rm(minimization)} \\
|
|
&&&&&& & d_k & \leq & 0\ {\rm(maximization)} \\
|
|
{\rm if} & -\infty & < & x_k & \leq & u_k,
|
|
& {\rm then} & d_k & \leq & 0\ {\rm(minimization)} \\
|
|
&&&&&& & d_k & \geq & 0\ {\rm(maximization)} \\
|
|
{\rm if} & l_k & \leq & x_k & \leq & u_k,
|
|
& {\rm then} & d_k & {\rm is} & {\rm of\ any\ sign} \\
|
|
\end{array}\right.\eqno{\rm(KKT.DB)}
|
|
$$
|
|
for all $k=1,\dots,m+n$, where $d_k$ is a reduced cost (Lagrange
|
|
multiplier) of auxiliary ($1\leq k\leq m$) or structural
|
|
($m+1\leq k\leq m+n$) variable $x_k$. Geometrically this condition
|
|
expresses the requirement that constraints of the original problem must
|
|
``hold'' the point preventing its movement along the anti-gradient (in
|
|
case of minimization) or the gradient (in case of maximization) of the
|
|
objective function. In case of basic solution reduced costs of all
|
|
basic variables are placed on their active (zero) bounds, so actually
|
|
the condition (KKT.DB) needs to be checked for non-basic variables
|
|
only. If the dual solution has sufficient accuracy, this condition
|
|
shows the dual feasibility of the solution.
|
|
|
|
To check the condition (KKT.DB) the routine computes a vector of
|
|
residuals:
|
|
$$
|
|
v_k = \left\{
|
|
\begin{array}{ll}
|
|
0, & {\rm if}\ d_k\ {\rm has\ correct\ sign} \\
|
|
|d_k|, & {\rm if}\ d_k\ {\rm has\ wrong\ sign} \\
|
|
\end{array}
|
|
\right.
|
|
$$
|
|
for all $k=1,\dots,m+n$, and determines components of this vector that
|
|
correspond to largest absolute and relative errors:
|
|
$${\tt ae\_max}=\max_{1\leq k\leq m+n}|v_k|,$$
|
|
$${\tt re\_max}=\max_{1\leq k\leq m+n}\frac{|v_k|}{1+|d_k - c_k|}.$$
|
|
|
|
Note that the complete set of Karush-Kuhn-Tucker optimality conditions
|
|
also includes the fifth, so called {\it complementary slackness
|
|
condition}, which expresses the requirement that at least either
|
|
a primal variable $x_k$ or its dual counterpart $d_k$ should be on its
|
|
bound for all $k=1,\dots,m+n$. Currently checking this condition is
|
|
not implemented yet.
|
|
|
|
\def\arraystretch{1}
|
|
|
|
%* eof *%
|