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1385 lines
52 KiB
1385 lines
52 KiB
%* glpk04.tex *%
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\chapter{Advanced API Routines}
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\section{Background}
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\label{basbgd}
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Using vector and matrix notations the LP problem (1.1)---(1.3) (see
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Section \ref{seclp}, page \pageref{seclp}) can be stated as follows:
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\noindent
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\hspace{.5in} minimize (or maximize)
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$$z=c^Tx_S+c_0\eqno(3.1)$$
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\hspace{.5in} subject to linear constraints
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$$x_R=Ax_S\eqno(3.2)$$
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\hspace{.5in} and bounds of variables
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$$
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\begin{array}{l@{\ }c@{\ }l@{\ }c@{\ }l}
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l_R&\leq&x_R&\leq&u_R\\
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l_S&\leq&x_S&\leq&u_S\\
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\end{array}\eqno(3.3)
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$$
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where:
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$x_R=(x_1,\dots,x_m)$ is the vector of auxiliary variables;
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$x_S=(x_{m+1},\dots,x_{m+n})$ is the vector of structural variables;
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$z$ is the objective function;
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$c=(c_1,\dots,c_n)$ is the vector of objective coefficients;
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$c_0$ is the constant term (``shift'') of the objective function;
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$A=(a_{11},\dots,a_{mn})$ is the constraint matrix;
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$l_R=(l_1,\dots,l_m)$ is the vector of lower bounds of auxiliary
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variables;
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$u_R=(u_1,\dots,u_m)$ is the vector of upper bounds of auxiliary
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variables;
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$l_S=(l_{m+1},\dots,l_{m+n})$ is the vector of lower bounds of
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structural variables;
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$u_S=(u_{m+1},\dots,u_{m+n})$ is the vector of upper bounds of
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structural variables.
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From the simplex method's standpoint there is no difference between
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auxiliary and structural variables. This allows combining all these
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variables into one vector that leads to the following problem
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statement:
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\noindent
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\hspace{.5in} minimize (or maximize)
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$$z=(0\ |\ c)^Tx+c_0\eqno(3.4)$$
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\hspace{.5in} subject to linear constraints
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$$(I\ |-\!A)x=0\eqno(3.5)$$
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\hspace{.5in} and bounds of variables
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$$l\leq x\leq u\eqno(3.6)$$
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where:
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$x=(x_R\ |\ x_S)$ is the $(m+n)$-vector of (all) variables;
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$(0\ |\ c)$ is the $(m+n)$-vector of objective
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coefficients;\footnote{Subvector 0 corresponds to objective
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coefficients at auxiliary variables.}
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$(I\ |-\!A)$ is the {\it augmented} constraint
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$m\times(m+n)$-matrix;\footnote{Note that due to auxiliary variables
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matrix $(I\ |-\!A)$ contains the unity submatrix and therefore has full
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rank. This means, in particular, that the system (3.5) has no linearly
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dependent constraints.}
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$l=(l_R\ |\ l_S)$ is the $(m+n)$-vector of lower bounds of (all)
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variables;
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$u=(u_R\ |\ u_S)$ is the $(m+n)$-vector of upper bounds of (all)
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variables.
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By definition an {\it LP basic solution} geometrically is a point in
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the space of all variables, which is the intersection of hyperplanes
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corresponding to active constraints\footnote{A constraint is called
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{\it active} if at a given point it is satisfied as equality, otherwise
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it is called {\it inactive}.}. The space of all variables has the
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dimension $m+n$, therefore, to define some basic solution we have to
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define $m+n$ active constraints. Note that $m$ constraints (3.5) being
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linearly independent equalities are always active, so remaining $n$
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active constraints can be chosen only from bound constraints (3.6).
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A variable is called {\it non-basic}, if its (lower or upper) bound is
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active, otherwise it is called {\it basic}. Since, as was said above,
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exactly $n$ bound constraints must be active, in any basic solution
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there are always $n$ non-basic variables and $m$ basic variables.
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(Note that a free variable also can be non-basic. Although such
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variable has no bounds, we can think it as the difference between two
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non-negative variables, which both are non-basic in this case.)
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Now consider how to determine numeric values of all variables for a
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given basic solution.
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Let $\Pi$ be an appropriate permutation matrix of the order $(m+n)$.
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Then we can write:
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$$\left(\begin{array}{@{}c@{}}x_B\\x_N\\\end{array}\right)=
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\Pi\left(\begin{array}{@{}c@{}}x_R\\x_S\\\end{array}\right)=\Pi x,
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\eqno(3.7)$$
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where $x_B$ is the vector of basic variables, $x_N$ is the vector of
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non-basic variables, $x=(x_R\ |\ x_S)$ is the vector of all variables
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in the original order. In this case the system of linear constraints
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(3.5) can be rewritten as follows:
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$$(I\ |-\!A)\Pi^T\Pi x=0\ \ \ \Rightarrow\ \ \ (B\ |\ N)
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\left(\begin{array}{@{}c@{}}x_B\\x_N\\\end{array}\right)=0,\eqno(3.8)$$
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where
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$$(B\ |\ N)=(I\ |-\!A)\Pi^T.\eqno(3.9)$$
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\newpage
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Matrix $B$ is a square non-singular $m\times m$-matrix, which is
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composed from columns of the augmented constraint matrix corresponding
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to basic variables. It is called the {\it basis matrix} or simply the
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{\it basis}. Matrix $N$ is a rectangular $m\times n$-matrix, which is
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composed from columns of the augmented constraint matrix corresponding
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to non-basic variables.
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From (3.8) it follows that:
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$$Bx_B+Nx_N=0,\eqno(3.10)$$
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therefore,
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$$x_B=-B^{-1}Nx_N.\eqno(3.11)$$
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Thus, the formula (3.11) shows how to determine numeric values of basic
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variables $x_B$ assuming that non-basic variables $x_N$ are fixed on
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their active bounds.
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The $m\times n$-matrix
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$$\Xi=-B^{-1}N,\eqno(3.12)$$
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which appears in (3.11), is called the {\it simplex
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tableau}.\footnote{This definition corresponds to the GLPK
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implementation.} It shows how basic variables depend on non-basic
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variables:
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$$x_B=\Xi x_N.\eqno(3.13)$$
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The system (3.13) is equivalent to the system (3.5) in the sense that
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they both define the same set of points in the space of (primal)
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variables, which satisfy to these systems. If, moreover, values of all
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basic variables satisfy to their bound constraints (3.3), the
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corresponding basic solution is called {\it (primal) feasible},
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otherwise {\it (primal) infeasible}. It is understood that any (primal)
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feasible basic solution satisfy to all constraints (3.2) and (3.3).
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The LP theory says that if LP has optimal solution, it has (at least
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one) basic feasible solution, which corresponds to the optimum. And the
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most natural way to determine whether a given basic solution is optimal
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or not is to use the Karush---Kuhn---Tucker optimality conditions.
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\def\arraystretch{1.5}
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For the problem statement (3.4)---(3.6) the optimality conditions are
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the following:\footnote{These conditions can be appiled to any solution,
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not only to a basic solution.}
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$$(I\ |-\!A)x=0\eqno(3.14)$$
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$$(I\ |-\!A)^T\pi+\lambda_l+\lambda_u=\nabla z=(0\ |\ c)^T\eqno(3.15)$$
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$$l\leq x\leq u\eqno(3.16)$$
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$$\lambda_l\geq 0,\ \ \lambda_u\leq 0\ \ \mbox{(minimization)}
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\eqno(3.17)$$
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$$\lambda_l\leq 0,\ \ \lambda_u\geq 0\ \ \mbox{(maximization)}
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\eqno(3.18)$$
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$$(\lambda_l)_k(x_k-l_k)=0,\ \ (\lambda_u)_k(x_k-u_k)=0,\ \ k=1,2,\dots,
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m+n\eqno(3.19)$$
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where:
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$\pi=(\pi_1,\pi_2,\dots,\pi_m)$ is a $m$-vector of Lagrange
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multipliers for equality constraints (3.5);
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$\lambda_l=[(\lambda_l)_1,(\lambda_l)_2,\dots,(\lambda_l)_n]$ is a
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$n$-vector of Lagrange multipliers for lower bound constraints (3.6);
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$\lambda_u=[(\lambda_u)_1,(\lambda_u)_2,\dots,(\lambda_u)_n]$ is a
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$n$-vector of Lagrange multipliers for upper bound constraints (3.6).
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\newpage
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Condition (3.14) is the {\it primal} (original) system of equality
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constraints (3.5).
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Condition (3.15) is the {\it dual} system of equality constraints.
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It requires the gradient of the objective function to be a linear
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combination of normals to the planes defined by constraints of the
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original problem.
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Condition (3.16) is the primal (original) system of bound constraints
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(3.6).
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Condition (3.17) (or (3.18) in case of maximization) is the dual system
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of bound constraints.
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Condition (3.19) is the {\it complementary slackness condition}. It
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requires, for each original (auxiliary or structural) variable $x_k$,
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that either its (lower or upper) bound must be active, or zero bound of
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the corresponding Lagrange multiplier ($(\lambda_l)_k$ or
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$(\lambda_u)_k$) must be active.
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In GLPK two multipliers $(\lambda_l)_k$ and $(\lambda_u)_k$ for each
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primal (original) variable $x_k$, $k=1,2,\dots,\linebreak m+n$, are
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combined into one multiplier:
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$$\lambda_k=(\lambda_l)_k+(\lambda_u)_k,\eqno(3.20)$$
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which is called a {\it dual variable} for $x_k$. This {\it cannot} lead
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to the ambiguity, because both lower and upper bounds of $x_k$ cannot be
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active at the same time,\footnote{If $x_k$ is a fixed variable, we can
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think it as double-bounded variable $l_k\leq x_k\leq u_k$, where
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$l_k=u_k.$} so at least one of $(\lambda_l)_k$ and $(\lambda_u)_k$ must
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be equal to zero, and because these multipliers have different signs,
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the combined multiplier, which is their sum, uniquely defines each of
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them.
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\def\arraystretch{1}
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Using dual variables $\lambda_k$ the dual system of bound constraints
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(3.17) and (3.18) can be written in the form of so called {\it ``rule of
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signs''} as follows:
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\medskip
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\begin{center}
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\begin{tabular}{|@{\,}c@{$\,$}|@{$\,$}c@{$\,$}|@{$\,$}c@{$\,$}|
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@{$\,$}c|c@{$\,$}|@{$\,$}c@{$\,$}|@{$\,$}c@{$\,$}|}
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\hline
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Original bound&\multicolumn{3}{c|}{Minimization}&\multicolumn{3}{c|}
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{Maximization}\\
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\cline{2-7}
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constraint&$(\lambda_l)_k$&$(\lambda_u)_k$&$(\lambda_l)_k+
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(\lambda_u)_k$&$(\lambda_l)_k$&$(\lambda_u)_k$&$(\lambda_l)_k+
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(\lambda_u)_k$\\
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\hline
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$-\infty<x_k<+\infty$&$=0$&$=0$&$\lambda_k=0$&$=0$&$=0$&$\lambda_k=0$\\
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$x_k\geq l_k$&$\geq 0$&$=0$&$\lambda_k\geq 0$&$\leq 0$&$=0$&$\lambda_k
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\leq0$\\
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$x_k\leq u_k$&$=0$&$\leq 0$&$\lambda_k\leq 0$&$=0$&$\geq 0$&$\lambda_k
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\geq0$\\
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$l_k\leq x_k\leq u_k$&$\geq 0$& $\leq 0$& $-\infty\!<\!\lambda_k\!<
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\!+\infty$
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&$\leq 0$& $\geq 0$& $-\infty\!<\!\lambda_k\!<\!+\infty$\\
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$x_k=l_k=u_k$&$\geq 0$& $\leq 0$& $-\infty\!<\!\lambda_k\!<\!+\infty$&
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$\leq 0$&
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$\geq 0$& $-\infty\!<\!\lambda_k\!<\!+\infty$\\
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\hline
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\end{tabular}
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\end{center}
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\medskip
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May note that each primal variable $x_k$ has its dual counterpart
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$\lambda_k$ and vice versa. This allows applying the same partition for
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the vector of dual variables as (3.7):
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$$\left(\begin{array}{@{}c@{}}\lambda_B\\\lambda_N\\\end{array}\right)=
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\Pi\lambda,\eqno(3.21)$$
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where $\lambda_B$ is a vector of dual variables for basic variables
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$x_B$, $\lambda_N$ is a vector of dual variables for non-basic variables
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$x_N$.
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By definition, bounds of basic variables are inactive constraints, so in
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any basic solution $\lambda_B=0$. Corresponding values of dual variables
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$\lambda_N$ for non-basic variables $x_N$ can be determined in the
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following way. From the dual system (3.15) we have:
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$$(I\ |-\!A)^T\pi+\lambda=(0\ |\ c)^T,\eqno(3.22)$$
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\newpage
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\noindent
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so multiplying both sides of (3.22) by matrix $\Pi$ gives:
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$$\Pi(I\ |-\!A)^T\pi+\Pi\lambda=\Pi(0\ |\ c)^T.\eqno(3.23)$$
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From (3.9) it follows that
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$$\Pi(I\ |-\!A)^T=[(I\ |-\!A)\Pi^T]^T=(B\ |\ N)^T.\eqno(3.24)$$
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Further, we can apply the partition (3.7) also to the vector of
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objective coefficients (see (3.4)):
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$$\left(\begin{array}{@{}c@{}}c_B\\c_N\\\end{array}\right)=
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\Pi\left(\begin{array}{@{}c@{}}0\\c\\\end{array}\right),\eqno(3.25)$$
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where $c_B$ is a vector of objective coefficients at basic variables,
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$c_N$ is a vector of objective coefficients at non-basic variables.
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Now, substituting (3.24), (3.21), and (3.25) into (3.23), leads to:
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$$(B\ |\ N)^T\pi+(\lambda_B\ |\ \lambda_N)^T=(c_B\ |\ c_N)^T,
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\eqno(3.26)$$
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and transposing both sides of (3.26) gives the system:
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$$\left(\begin{array}{@{}c@{}}B^T\\N^T\\\end{array}\right)\pi+
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\left(\begin{array}{@{}c@{}}\lambda_B\\\lambda_N\\\end{array}\right)=
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\left(\begin{array}{@{}c@{}}c_B\\c_T\\\end{array}\right),\eqno(3.27)$$
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which can be written as follows:
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$$\left\{
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\begin{array}{@{\ }r@{\ }c@{\ }r@{\ }c@{\ }l@{\ }}
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B^T\pi&+&\lambda_B&=&c_B\\
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N^T\pi&+&\lambda_N&=&c_N\\
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\end{array}
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\right.\eqno(3.28)
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$$
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Lagrange multipliers $\pi=(\pi_i)$ correspond to equality constraints
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(3.5) and therefore can have any sign. This allows resolving the first
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subsystem of (3.28) as follows:\footnote{$B^{-T}$ means $(B^T)^{-1}=
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(B^{-1})^T$.}
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$$\pi=B^{-T}(c_B-\lambda_B)=-B^{-T}\lambda_B+B^{-T}c_B,\eqno(3.29)$$
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and substitution of $\pi$ from (3.29) into the second subsystem of
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(3.28) gives:
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$$\lambda_N=-N^T\pi+c_N=N^TB^{-T}\lambda_B+(c_N-N^TB^{-T}c_B).
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\eqno(3.30)$$
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The latter system can be written in the following final form:
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$$\lambda_N=-\Xi^T\lambda_B+d,\eqno(3.31)$$
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where $\Xi$ is the simplex tableau (see (3.12)), and
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$$d=c_N-N^TB^{-T}c_B=c_N+\Xi^Tc_B\eqno(3.32)$$
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is the vector of so called {\it reduced costs} of non-basic variables.
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Above it was said that in any basic solution $\lambda_B=0$, so
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$\lambda_N=d$ as it follows from (3.31).
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The system (3.31) is equivalent to the system (3.15) in the sense that
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they both define the same set of points in the space of dual variables
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$\lambda$, which satisfy to these systems. If, moreover, values of all
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dual variables $\lambda_N$ (i.e. reduced costs $d$) satisfy to their
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bound constraints (i.e. to the ``rule of signs''; see the table above),
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the corresponding basic solution is called {\it dual feasible},
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otherwise {\it dual infeasible}. It is understood that any dual feasible
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solution satisfy to all constraints (3.15) and (3.17) (or (3.18) in case
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of maximization).
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It can be easily shown that the complementary slackness condition
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(3.19) is always satisfied for {\it any} basic solution.\footnote{Until
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double-bounded variables appear.} Therefore, a basic
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solution\footnote{It is assumed that a complete basic solution has the
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form $(x,\lambda)$, i.e. it includes primal as well as dual variables.}
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is {\it optimal} if and only if it is primal and dual feasible, because
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in this case it satifies to all the optimality conditions
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(3.14)---(3.19).
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\def\arraystretch{1.5}
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The meaning of reduced costs $d=(d_j)$ of non-basic variables can be
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explained in the following way. From (3.4), (3.7), and (3.25) it follows
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that:
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$$z=c_B^Tx_B+c_N^Tx_N+c_0.\eqno(3.33)$$
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Substituting $x_B$ from (3.11) into (3.33) we can eliminate basic
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variables and express the objective only through non-basic variables:
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$$
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\begin{array}{r@{\ }c@{\ }l}
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z&=&c_B^T(-B^{-1}Nx_N)+c_N^Tx_N+c_0=\\
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&=&(c_N^T-c_B^TB^{-1}N)x_N+c_0=\\
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&=&(c_N-N^TB^{-T}c_B)^Tx_N+c_0=\\
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&=&d^Tx_N+c_0.
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\end{array}\eqno(3.34)
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$$
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From (3.34) it is seen that reduced cost $d_j$ shows how the objective
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function $z$ depends on non-basic variable $(x_N)_j$ in the neighborhood
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of the current basic solution, i.e. while the current basis remains
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unchanged.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newpage
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\section{LP basis routines}
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\label{lpbasis}
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\subsection{glp\_bf\_exists --- check if the basis factorization
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exists}
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\synopsis
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\begin{verbatim}
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int glp_bf_exists(glp_prob *P);
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\end{verbatim}
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\returns
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If the basis factorization for the current basis associated with the
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specified problem object exists and therefore is available for
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computations, the routine \verb|glp_bf_exists| returns non-zero.
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Otherwise the routine returns zero.
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\para{Comments}
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Let the problem object have $m$ rows and $n$ columns. In GLPK the
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{\it basis matrix} $B$ is a square non-singular matrix of the order $m$,
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whose columns correspond to basic (auxiliary and/or structural)
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variables. It is defined by the following main
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equality:\footnote{For more details see Subsection \ref{basbgd},
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page \pageref{basbgd}.}
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$$(B\ |\ N)=(I\ |-\!A)\Pi^T,$$
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where $I$ is the unity matrix of the order $m$, whose columns correspond
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to auxiliary variables; $A$ is the original constraint
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$m\times n$-matrix, whose columns correspond to structural variables;
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$(I\ |-\!A)$ is the augmented constraint $m\times(m+n)$-matrix, whose
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columns correspond to all (auxiliary and structural) variables
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following in the original order; $\Pi$ is a permutation matrix of the
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order $m+n$; and $N$ is a rectangular $m\times n$-matrix, whose columns
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correspond to non-basic (auxiliary and/or structural) variables.
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For various reasons it may be necessary to solve linear systems with
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matrix $B$. To provide this possibility the GLPK implementation
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maintains an invertable form of $B$ (that is, some representation of
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$B^{-1}$) called the {\it basis factorization}, which is an internal
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component of the problem object. Typically, the basis factorization is
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computed by the simplex solver, which keeps it in the problem object
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to be available for other computations.
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Should note that any changes in the problem object, which affects the
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basis matrix (e.g. changing the status of a row or column, changing
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a basic column of the constraint matrix, removing an active constraint,
|
|
etc.), invalidates the basis factorization. So before calling any API
|
|
routine, which uses the basis factorization, the application program
|
|
must make sure (using the routine \verb|glp_bf_exists|) that the
|
|
factorization exists and therefore available for computations.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_factorize --- compute the basis factorization}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_factorize(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_factorize| computes the basis factorization for
|
|
the current basis associated with the specified problem
|
|
object.\footnote{The current basis is defined by the current statuses
|
|
of rows (auxiliary variables) and columns (structural variables).}
|
|
|
|
The basis factorization is computed from ``scratch'' even if it exists,
|
|
so the application program may use the routine \verb|glp_bf_exists|,
|
|
and, if the basis factorization already exists, not to call the routine
|
|
\verb|glp_factorize| to prevent an extra work.
|
|
|
|
The routine \verb|glp_factorize| {\it does not} compute components of
|
|
the basic solution (i.e. primal and dual values).
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The basis factorization has been successfully computed.\\
|
|
\verb|GLP_EBADB| & The basis matrix is invalid, because the number of
|
|
basic (auxiliary and structural) variables is not the same as the number
|
|
of rows in the problem object.\\
|
|
|
|
\verb|GLP_ESING| & The basis matrix is singular within the working
|
|
precision.\\
|
|
|
|
\verb|GLP_ECOND| & The basis matrix is ill-conditioned, i.e. its
|
|
condition number is too large.\\
|
|
\end{retlist}
|
|
|
|
\subsection{glp\_bf\_updated --- check if the basis factorization has
|
|
been updated}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_bf_updated(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
If the basis factorization has been just computed from ``scratch'', the
|
|
routine \verb|glp_bf_updated| returns zero. Otherwise, if the
|
|
factorization has been updated at least once, the routine returns
|
|
non-zero.
|
|
|
|
\para{Comments}
|
|
|
|
{\it Updating} the basis factorization means recomputing it to reflect
|
|
changes in the basis matrix. For example, on every iteration of the
|
|
simplex method some column of the current basis matrix is replaced by
|
|
a new column that gives a new basis matrix corresponding to the
|
|
adjacent basis. In this case computing the basis factorization for the
|
|
adjacent basis from ``scratch'' (as the routine \verb|glp_factorize|
|
|
does) would be too time-consuming.
|
|
|
|
On the other hand, since the basis factorization update is a numeric
|
|
computational procedure, applying it many times may lead to
|
|
accumulating round-off errors. Therefore the basis is periodically
|
|
refactorized (reinverted) from ``scratch'' (with the routine
|
|
\verb|glp_factorize|) that allows improving its numerical properties.
|
|
|
|
The routine \verb|glp_bf_updated| allows determining if the basis
|
|
factorization has been updated at least once since it was computed from
|
|
``scratch''.
|
|
|
|
\subsection{glp\_get\_bfcp --- retrieve basis factorization control
|
|
parameters}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_get_bfcp(glp_prob *P, glp_bfcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_get_bfcp| retrieves control parameters, which are
|
|
used on computing and updating the basis factorization associated with
|
|
the specified problem object.
|
|
|
|
Current values of the control parameters are stored in
|
|
a \verb|glp_bfcp| structure, which the parameter \verb|parm| points to.
|
|
For a detailed description of the structure \verb|glp_bfcp| see
|
|
comments to the routine \verb|glp_set_bfcp| in the next subsection.
|
|
|
|
\para{Comments}
|
|
|
|
The purpose of the routine \verb|glp_get_bfcp| is two-fold. First, it
|
|
allows the application program obtaining current values of control
|
|
parameters used by internal GLPK routines, which compute and update the
|
|
basis factorization.
|
|
|
|
The second purpose of this routine is to provide proper values for all
|
|
fields of the structure \verb|glp_bfcp| in the case when the
|
|
application program needs to change some control parameters.
|
|
|
|
\subsection{glp\_set\_bfcp --- change basis factorization control
|
|
parameters}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_set_bfcp(glp_prob *P, const glp_bfcp *parm);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_set_bfcp| changes control parameters, which are
|
|
used by internal GLPK routines on computing and updating the basis
|
|
factorization associated with the specified problem object.
|
|
|
|
New values of the control parameters should be passed in a structure
|
|
\verb|glp_bfcp|, which the parameter \verb|parm| points to. For a
|
|
detailed description of the structure \verb|glp_bfcp| see paragraph
|
|
``Control parameters'' below.
|
|
|
|
The parameter \verb|parm| can be specified as \verb|NULL|, in which
|
|
case all control parameters are reset to their default values.
|
|
|
|
\para{Comments}
|
|
|
|
Before changing some control parameters with the routine
|
|
\verb|glp_set_bfcp| the application program should retrieve current
|
|
values of all control parameters with the routine \verb|glp_get_bfcp|.
|
|
This is needed for backward compatibility, because in the future there
|
|
may appear new members in the structure \verb|glp_bfcp|.
|
|
|
|
Note that new values of control parameters come into effect on a next
|
|
computation of the basis factorization, not immediately.
|
|
|
|
\para{Example}
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
glp_prob *lp;
|
|
glp_bfcp parm;
|
|
. . .
|
|
/* retrieve current values of control parameters */
|
|
glp_get_bfcp(lp, &parm);
|
|
/* change the threshold pivoting tolerance */
|
|
parm.piv_tol = 0.05;
|
|
/* set new values of control parameters */
|
|
glp_set_bfcp(lp, &parm);
|
|
. . .
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
\para{Control parameters}
|
|
|
|
This paragraph describes all basis factorization control parameters
|
|
currently used in the package. Symbolic names of control parameters are
|
|
names of corresponding members in the structure \verb|glp_bfcp|.
|
|
|
|
\medskip
|
|
|
|
{\tt int type} (default: {\tt GLP\_BF\_FT})
|
|
|
|
Basis factorization type:
|
|
|
|
\verb|GLP_BF_FT|---$LU$ + Forrest--Tomlin update;
|
|
|
|
\verb|GLP_BF_BG|---$LU$ + Schur complement + Bartels--Golub update;
|
|
|
|
\verb|GLP_BF_GR|---$LU$ + Schur complement + Givens rotation update.
|
|
|
|
In case of \verb|GLP_BF_FT| the update is applied to matrix $U$, while
|
|
in cases of \verb|GLP_BF_BG| and \verb|GLP_BF_GR| the update is applied
|
|
to the Schur complement.
|
|
|
|
\medskip
|
|
|
|
{\tt int lu\_size} (default: {\tt 0})
|
|
|
|
The initial size of the Sparse Vector Area, in non-zeros, used on
|
|
computing $LU$-factorization of the basis matrix for the first time.
|
|
If this parameter is set to 0, the initial SVA size is determined
|
|
automatically.
|
|
|
|
\medskip
|
|
|
|
{\tt double piv\_tol} (default: {\tt 0.10})
|
|
|
|
Threshold pivoting (Markowitz) tolerance, 0 $<$ \verb|piv_tol| $<$ 1,
|
|
used on computing $LU$-factoriza\-tion of the basis matrix. Element
|
|
$u_{ij}$ of the active submatrix of factor $U$ fits to be pivot if it
|
|
satisfies to the stability criterion
|
|
$|u_{ij}| >= {\tt piv\_tol}\cdot\max|u_{i*}|$, i.e. if it is not very
|
|
small in the magnitude among other elements in the same row. Decreasing
|
|
this parameter may lead to better sparsity at the expense of numerical
|
|
accuracy, and vice versa.
|
|
|
|
\medskip
|
|
|
|
{\tt int piv\_lim} (default: {\tt 4})
|
|
|
|
This parameter is used on computing $LU$-factorization of the basis
|
|
matrix and specifies how many pivot candidates needs to be considered
|
|
on choosing a pivot element, \verb|piv_lim| $\geq$ 1. If \verb|piv_lim|
|
|
candidates have been considered, the pivoting routine prematurely
|
|
terminates the search with the best candidate found.
|
|
|
|
\newpage
|
|
|
|
{\tt int suhl} (default: {\tt GLP\_ON})
|
|
|
|
This parameter is used on computing $LU$-factorization of the basis
|
|
matrix. Being set to {\tt GLP\_ON} it enables applying the following
|
|
heuristic proposed by Uwe Suhl: if a column of the active submatrix has
|
|
no eligible pivot candidates, it is no more considered until it becomes
|
|
a column singleton. In many cases this allows reducing the time needed
|
|
for pivot searching. To disable this heuristic the parameter
|
|
\verb|suhl| should be set to {\tt GLP\_OFF}.
|
|
|
|
\medskip
|
|
|
|
{\tt double eps\_tol} (default: {\tt 1e-15})
|
|
|
|
Epsilon tolerance, \verb|eps_tol| $\geq$ 0, used on computing
|
|
$LU$-factorization of the basis matrix. If an element of the active
|
|
submatrix of factor $U$ is less than \verb|eps_tol| in the magnitude,
|
|
it is replaced by exact zero.
|
|
|
|
\medskip
|
|
|
|
{\tt double max\_gro} (default: {\tt 1e+10})
|
|
|
|
Maximal growth of elements of factor $U$, \verb|max_gro| $\geq$ 1,
|
|
allowable on computing $LU$-factorization of the basis matrix. If on
|
|
some elimination step the ratio $u_{big}/b_{max}$ (where $u_{big}$ is
|
|
the largest magnitude of elements of factor $U$ appeared in its active
|
|
submatrix during all the factorization process, $b_{max}$ is the
|
|
largest magnitude of elements of the basis matrix to be factorized),
|
|
the basis matrix is considered as ill-conditioned.
|
|
|
|
\medskip
|
|
|
|
{\tt int nfs\_max} (default: {\tt 100})
|
|
|
|
Maximal number of additional row-like factors (entries of the eta
|
|
file), \verb|nfs_max| $\geq$ 1, which can be added to
|
|
$LU$-factorization of the basis matrix on updating it with the
|
|
Forrest--Tomlin technique. This parameter is used only once, before
|
|
$LU$-factorization is computed for the first time, to allocate working
|
|
arrays. As a rule, each update adds one new factor (however, some
|
|
updates may need no addition), so this parameter limits the number of
|
|
updates between refactorizations.
|
|
|
|
\medskip
|
|
|
|
{\tt double upd\_tol} (default: {\tt 1e-6})
|
|
|
|
Update tolerance, 0 $<$ \verb|upd_tol| $<$ 1, used on updating
|
|
$LU$-factorization of the basis matrix with the Forrest--Tomlin
|
|
technique. If after updating the magnitude of some diagonal element
|
|
$u_{kk}$ of factor $U$ becomes less than
|
|
${\tt upd\_tol}\cdot\max(|u_{k*}|, |u_{*k}|)$, the factorization is
|
|
considered as inaccurate.
|
|
|
|
\medskip
|
|
|
|
{\tt int nrs\_max} (default: {\tt 100})
|
|
|
|
Maximal number of additional rows and columns, \verb|nrs_max| $\geq$ 1,
|
|
which can be added to $LU$-factorization of the basis matrix on
|
|
updating it with the Schur complement technique. This parameter is used
|
|
only once, before $LU$-factorization is computed for the first time, to
|
|
allocate working arrays. As a rule, each update adds one new row and
|
|
column (however, some updates may need no addition), so this parameter
|
|
limits the number of updates between refactorizations.
|
|
|
|
\medskip
|
|
|
|
{\tt int rs\_size} (default: {\tt 0})
|
|
|
|
The initial size of the Sparse Vector Area, in non-zeros, used to
|
|
store non-zero elements of additional rows and columns introduced on
|
|
updating $LU$-factorization of the basis matrix with the Schur
|
|
complement technique. If this parameter is set to 0, the initial SVA
|
|
size is determined automatically.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_get\_bhead --- retrieve the basis header information}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_bhead(glp_prob *P, int k);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_get_bhead| returns the basis header information
|
|
for the current basis associated with the specified problem object.
|
|
|
|
\returns
|
|
|
|
If basic variable $(x_B)_k$, $1\leq k\leq m$, is $i$-th auxiliary
|
|
variable ($1\leq i\leq m$), the routine returns $i$. Otherwise, if
|
|
$(x_B)_k$ is $j$-th structural variable ($1\leq j\leq n$), the routine
|
|
returns $m+j$. Here $m$ is the number of rows and $n$ is the number of
|
|
columns in the problem object.
|
|
|
|
\para{Comments}
|
|
|
|
Sometimes the application program may need to know which original
|
|
(auxiliary and structural) variable correspond to a given basic
|
|
variable, or, that is the same, which column of the augmented
|
|
constraint matrix $(I\ |-\!A)$ correspond to a given column of the
|
|
basis matrix $B$.
|
|
|
|
\def\arraystretch{1}
|
|
|
|
The correspondence is defined as follows:\footnote{For more details see
|
|
Subsection \ref{basbgd}, page \pageref{basbgd}.}
|
|
$$\left(\begin{array}{@{}c@{}}x_B\\x_N\\\end{array}\right)=
|
|
\Pi\left(\begin{array}{@{}c@{}}x_R\\x_S\\\end{array}\right)
|
|
\ \ \Leftrightarrow
|
|
\ \ \left(\begin{array}{@{}c@{}}x_R\\x_S\\\end{array}\right)=
|
|
\Pi^T\left(\begin{array}{@{}c@{}}x_B\\x_N\\\end{array}\right),$$
|
|
where $x_B$ is the vector of basic variables, $x_N$ is the vector of
|
|
non-basic variables, $x_R$ is the vector of auxiliary variables
|
|
following in their original order,\footnote{The original order of
|
|
auxiliary and structural variables is defined by the ordinal numbers
|
|
of corresponding rows and columns in the problem object.} $x_S$ is the
|
|
vector of structural variables following in their original order, $\Pi$
|
|
is a permutation matrix (which is a component of the basis
|
|
factorization).
|
|
|
|
Thus, if $(x_B)_k=(x_R)_i$ is $i$-th auxiliary variable, the routine
|
|
returns $i$, and if $(x_B)_k=(x_S)_j$ is $j$-th structural variable,
|
|
the routine returns $m+j$, where $m$ is the number of rows in the
|
|
problem object.
|
|
|
|
\subsection{glp\_get\_row\_bind --- retrieve row index in the basis
|
|
header}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_row_bind(glp_prob *P, int i);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_row_bind| returns the index $k$ of basic
|
|
variable $(x_B)_k$, $1\leq k\leq m$, which is $i$-th auxiliary variable
|
|
(that is, the auxiliary variable corresponding to $i$-th row),
|
|
$1\leq i\leq m$, in the current basis associated with the specified
|
|
problem object, where $m$ is the number of rows. However, if $i$-th
|
|
auxiliary variable is non-basic, the routine returns zero.
|
|
|
|
\newpage
|
|
|
|
\para{Comments}
|
|
|
|
The routine \verb|glp_get_row_bind| is an inversion of the routine
|
|
\verb|glp_get_bhead|; that is, if \linebreak
|
|
\verb|glp_get_bhead|$(P,k)$ returns $i$,
|
|
\verb|glp_get_row_bind|$(P,i)$ returns $k$, and vice versa.
|
|
|
|
\subsection{glp\_get\_col\_bind --- retrieve column index in the basis
|
|
header}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_get_col_bind(glp_prob *P, int j);
|
|
\end{verbatim}
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_get_col_bind| returns the index $k$ of basic
|
|
variable $(x_B)_k$, $1\leq k\leq m$, which is $j$-th structural
|
|
variable (that is, the structural variable corresponding to $j$-th
|
|
column), $1\leq j\leq n$, in the current basis associated with the
|
|
specified problem object, where $m$ is the number of rows, $n$ is the
|
|
number of columns. However, if $j$-th structural variable is non-basic,
|
|
the routine returns zero.
|
|
|
|
\para{Comments}
|
|
|
|
The routine \verb|glp_get_col_bind| is an inversion of the routine
|
|
\verb|glp_get_bhead|; that is, if \linebreak
|
|
\verb|glp_get_bhead|$(P,k)$ returns $m+j$,
|
|
\verb|glp_get_col_bind|$(P,j)$ returns $k$, and vice versa.
|
|
|
|
\subsection{glp\_ftran --- perform forward transformation}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_ftran(glp_prob *P, double x[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_ftran| performs forward transformation (FTRAN),
|
|
i.e. it solves the system $Bx=b$, where $B$ is the basis matrix
|
|
associated with the specified problem object, $x$ is the vector of
|
|
unknowns to be computed, $b$ is the vector of right-hand sides.
|
|
|
|
On entry to the routine elements of the vector $b$ should be stored in
|
|
locations \verb|x[1]|, \dots, \verb|x[m]|, where $m$ is the number of
|
|
rows. On exit the routine stores elements of the vector $x$ in the same
|
|
locations.
|
|
|
|
\subsection{glp\_btran --- perform backward transformation}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_btran(glp_prob *P, double x[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_btran| performs backward transformation (BTRAN),
|
|
i.e. it solves the system $B^Tx=b$, where $B^T$ is a matrix transposed
|
|
to the basis matrix $B$ associated with the specified problem object,
|
|
$x$ is the vector of unknowns to be computed, $b$ is the vector of
|
|
right-hand sides.
|
|
|
|
\newpage
|
|
|
|
On entry to the routine elements of the vector $b$ should be stored in
|
|
locations \verb|x[1]|, \dots, \verb|x[m]|, where $m$ is the number of
|
|
rows. On exit the routine stores elements of the vector $x$ in the same
|
|
locations.
|
|
|
|
\subsection{glp\_warm\_up --- ``warm up'' LP basis}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_warm_up(glp_prob *P);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_warm_up| ``warms up'' the LP basis for the
|
|
specified problem object using current statuses assigned to rows and
|
|
columns (that is, to auxiliary and structural variables).
|
|
|
|
This operation includes computing factorization of the basis matrix
|
|
(if it does not exist), computing primal and dual components of basic
|
|
solution, and determining the solution status.
|
|
|
|
\returns
|
|
|
|
\begin{retlist}
|
|
0 & The operation has been successfully performed.\\
|
|
|
|
\verb|GLP_EBADB| & The basis matrix is invalid, because the number of
|
|
basic (auxiliary and structural) variables is not the same as the
|
|
number of rows in the problem object.\\
|
|
|
|
\verb|GLP_ESING| & The basis matrix is singular within the working
|
|
precision.\\
|
|
|
|
\verb|GLP_ECOND| & The basis matrix is ill-conditioned, i.e. its
|
|
condition number is too large.\\
|
|
\end{retlist}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Simplex tableau routines}
|
|
|
|
\subsection{glp\_eval\_tab\_row --- compute row of the tableau}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_eval_tab_row(glp_prob *P, int k, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_eval_tab_row| computes a row of the current
|
|
simplex tableau (see Subsection 3.1.1, formula (3.12)), which (row)
|
|
corresponds to some basic variable specified by the parameter $k$ as
|
|
follows: if $1\leq k\leq m$, the basic variable is $k$-th auxiliary
|
|
variable, and if $m+1\leq k\leq m+n$, the basic variable is $(k-m)$-th
|
|
structural variable, where $m$ is the number of rows and $n$ is the
|
|
number of columns in the specified problem object. The basis
|
|
factorization must exist.
|
|
|
|
The computed row shows how the specified basic variable depends on
|
|
non-basic variables:
|
|
$$x_k=(x_B)_i=\xi_{i1}(x_N)_1+\xi_{i2}(x_N)_2+\dots+\xi_{in}(x_N)_n,$$
|
|
where $\xi_{i1}$, $\xi_{i2}$, \dots, $\xi_{in}$ are elements of the
|
|
simplex table row, $(x_N)_1$, $(x_N)_2$, \dots, $(x_N)_n$ are non-basic
|
|
(auxiliary and structural) variables.
|
|
|
|
The routine stores column indices and corresponding numeric values of
|
|
non-zero elements of the computed row in unordered sparse format in
|
|
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 non-zero elements in the row returned on exit.
|
|
|
|
Element indices stored in the array \verb|ind| have the same sense as
|
|
index $k$, i.e. indices 1 to $m$ denote auxiliary variables while
|
|
indices $m+1$ to $m+n$ denote structural variables (all these variables
|
|
are obviously non-basic by definition).
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_eval_tab_row| returns \verb|len|, which is the
|
|
number of non-zero elements in the simplex table row stored in the
|
|
arrays \verb|ind| and \verb|val|.
|
|
|
|
\para{Comments}
|
|
|
|
A row of the simplex table is computed as follows. At first, the
|
|
routine checks that the specified variable $x_k$ is basic and uses the
|
|
permutation matrix $\Pi$ (3.7) to determine index $i$ of basic variable
|
|
$(x_B)_i$, which corresponds to $x_k$.
|
|
|
|
The row to be computed is $i$-th row of the matrix $\Xi$ (3.12),
|
|
therefore:
|
|
$$\xi_i=e_i^T\Xi=-e_i^TB^{-1}N=-(B^{-T}e_i)^TN,$$
|
|
where $e_i$ is $i$-th unity vector. So the routine performs BTRAN to
|
|
obtain $i$-th row of the inverse $B^{-1}$:
|
|
$$\varrho_i=B^{-T}e_i,$$
|
|
and then computes elements of the simplex table row as inner products:
|
|
$$\xi_{ij}=-\varrho_i^TN_j,\ \ j=1,2,\dots,n,$$
|
|
where $N_j$ is $j$-th column of matrix $N$ (3.9), which (column)
|
|
corresponds to non-basic variable $(x_N)_j$. The permutation matrix
|
|
$\Pi$ is used again to convert indices $j$ of non-basic columns to
|
|
original ordinal numbers of auxiliary and structural variables.
|
|
|
|
\subsection{glp\_eval\_tab\_col --- compute column of the tableau}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_eval_tab_col(glp_prob *P, int k, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_eval_tab_col| computes a column of the current
|
|
simplex tableau (see Subsection 3.1.1, formula (3.12)), which (column)
|
|
corresponds to some non-basic variable specified by the parameter $k$:
|
|
if $1\leq k\leq m$, the non-basic variable is $k$-th auxiliary
|
|
variable, and if $m+1\leq k\leq m+n$, the non-basic variable is
|
|
$(k-m)$-th structural variable, where $m$ is the number of rows and $n$
|
|
is the number of columns in the specified problem object. The basis
|
|
factorization must exist.
|
|
|
|
The computed column shows how basic variables depends on the specified
|
|
non-basic variable $x_k=(x_N)_j$:
|
|
$$
|
|
\begin{array}{r@{\ }c@{\ }l@{\ }l}
|
|
(x_B)_1&=&\dots+\xi_{1j}(x_N)_j&+\dots\\
|
|
(x_B)_2&=&\dots+\xi_{2j}(x_N)_j&+\dots\\
|
|
.\ \ .&.&.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\\
|
|
(x_B)_m&=&\dots+\xi_{mj}(x_N)_j&+\dots\\
|
|
\end{array}
|
|
$$
|
|
where $\xi_{1j}$, $\xi_{2j}$, \dots, $\xi_{mj}$ are elements of the
|
|
simplex table column, $(x_B)_1$, $(x_B)_2$, \dots, $(x_B)_m$ are basic
|
|
(auxiliary and structural) variables.
|
|
|
|
The routine stores row indices and corresponding numeric values of
|
|
non-zero elements of the computed column in unordered sparse format in
|
|
locations \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 non-zero elements in the column returned on exit.
|
|
|
|
Element indices stored in the array \verb|ind| have the same sense as
|
|
index $k$, i.e. indices 1 to $m$ denote auxiliary variables while
|
|
indices $m+1$ to $m+n$ denote structural variables (all these variables
|
|
are obviously basic by definition).
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_eval_tab_col| returns \verb|len|, which is the
|
|
number of non-zero elements in the simplex table column stored in the
|
|
arrays \verb|ind| and \verb|val|.
|
|
|
|
\para{Comments}
|
|
|
|
A column of the simplex table is computed as follows. At first, the
|
|
routine checks that the specified variable $x_k$ is non-basic and uses
|
|
the permutation matrix $\Pi$ (3.7) to determine index $j$ of non-basic
|
|
variable $(x_N)_j$, which corresponds to $x_k$.
|
|
|
|
The column to be computed is $j$-th column of the matrix $\Xi$ (3.12),
|
|
therefore:
|
|
$$\Xi_j=\Xi e_j=-B^{-1}Ne_j=-B^{-1}N_j,$$
|
|
where $e_j$ is $j$-th unity vector, $N_j$ is $j$-th column of matrix
|
|
$N$ (3.9). So the routine performs FTRAN to transform $N_j$ to the
|
|
simplex table column $\Xi_j=(\xi_{ij})$ and uses the permutation matrix
|
|
$\Pi$ to convert row indices $i$ to original ordinal numbers of
|
|
auxiliary and structural variables.
|
|
|
|
\subsection{glp\_transform\_row --- transform explicitly specified row}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_transform_row(glp_prob *P, int len, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_transform_row| performs the same operation as the
|
|
routine \verb|glp_eval_tab_row| with exception that the row to be
|
|
transformed is specified explicitly as a sparse vector.
|
|
|
|
The explicitly specified row may be thought as a linear form:
|
|
$$x=a_1x_{m+1}+a_2x_{m+2}+\dots+a_nx_{m+n},$$
|
|
where $x$ is an auxiliary variable for this row, $a_j$ are coefficients
|
|
of the linear form, $x_{m+j}$ are structural variables.
|
|
|
|
On entry column indices and numerical values of non-zero coefficients
|
|
$a_j$ of the specified row should be placed in locations \verb|ind[1]|,
|
|
\dots, \verb|ind[len]| and \verb|val[1]|, \dots, \verb|val[len]|, where
|
|
\verb|len| is number of non-zero coefficients.
|
|
|
|
This routine uses the system of equality constraints and the current
|
|
basis in order to express the auxiliary variable $x$ through the current
|
|
non-basic variables (as if the transformed row were added to the problem
|
|
object and the auxiliary variable $x$ were basic), i.e. the resultant
|
|
row has the form:
|
|
$$x=\xi_1(x_N)_1+\xi_2(x_N)_2+\dots+\xi_n(x_N)_n,$$
|
|
where $\xi_j$ are influence coefficients, $(x_N)_j$ are non-basic
|
|
(auxiliary and structural) variables, $n$ is the number of columns in
|
|
the problem object.
|
|
|
|
On exit the routine stores indices and numerical values of non-zero
|
|
coefficients $\xi_j$ of the resultant row in locations \verb|ind[1]|,
|
|
\dots, \verb|ind[len']| and \verb|val[1]|, \dots, \verb|val[len']|,
|
|
where $0\leq{\tt len'}\leq n$ is the number of non-zero coefficients in
|
|
the resultant row returned by the routine. Note that indices of
|
|
non-basic variables stored in the array \verb|ind| correspond to
|
|
original ordinal numbers of variables: indices 1 to $m$ mean auxiliary
|
|
variables and indices $m+1$ to $m+n$ mean structural ones.
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_transform_row| returns \verb|len'|, the number of
|
|
non-zero coefficients in the resultant row stored in the arrays
|
|
\verb|ind| and \verb|val|.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_transform\_col --- transform explicitly specified
|
|
column}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_transform_col(glp_prob *P, int len, int ind[], double val[]);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_transform_col| performs the same operation as the
|
|
routine \verb|glp_eval_tab_col| with exception that the column to be
|
|
transformed is specified explicitly as a sparse vector.
|
|
|
|
The explicitly specified column may be thought as it were added to
|
|
the original system of equality constraints:
|
|
$$
|
|
\begin{array}{l@{\ }c@{\ }r@{\ }c@{\ }r@{\ }c@{\ }r}
|
|
x_1&=&a_{11}x_{m+1}&+\dots+&a_{1n}x_{m+n}&+&a_1x \\
|
|
x_2&=&a_{21}x_{m+1}&+\dots+&a_{2n}x_{m+n}&+&a_2x \\
|
|
\multicolumn{7}{c}
|
|
{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}\\
|
|
x_m&=&a_{m1}x_{m+1}&+\dots+&a_{mn}x_{m+n}&+&a_mx \\
|
|
\end{array}
|
|
$$
|
|
where $x_i$ are auxiliary variables, $x_{m+j}$ are structural variables
|
|
(presented in the problem object), $x$ is a structural variable for the
|
|
explicitly specified column, $a_i$ are constraint coefficients at $x$.
|
|
|
|
On entry row indices and numerical values of non-zero coefficients
|
|
$a_i$ of the specified column should be placed in locations
|
|
\verb|ind[1]|, \dots, \verb|ind[len]| and \verb|val[1]|, \dots,
|
|
\verb|val[len]|, where \verb|len| is number of non-zero coefficients.
|
|
|
|
This routine uses the system of equality constraints and the current
|
|
basis in order to express the current basic variables through the
|
|
structural variable $x$ (as if the transformed column were added to the
|
|
problem object and the variable $x$ were non-basic):
|
|
$$
|
|
\begin{array}{l@{\ }c@{\ }r}
|
|
(x_B)_1&=\dots+&\xi_{1}x\\
|
|
(x_B)_2&=\dots+&\xi_{2}x\\
|
|
\multicolumn{3}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .}\\
|
|
(x_B)_m&=\dots+&\xi_{m}x\\
|
|
\end{array}
|
|
$$
|
|
where $\xi_i$ are influence coefficients, $x_B$ are basic (auxiliary
|
|
and structural) variables, $m$ is the number of rows in the problem
|
|
object.
|
|
|
|
On exit the routine stores indices and numerical values of non-zero
|
|
coefficients $\xi_i$ of the resultant column in locations \verb|ind[1]|,
|
|
\dots, \verb|ind[len']| and \verb|val[1]|, \dots, \verb|val[len']|,
|
|
where $0\leq{\tt len'}\leq m$ is the number of non-zero coefficients in
|
|
the resultant column returned by the routine. Note that indices of basic
|
|
variables stored in the array \verb|ind| correspond to original ordinal
|
|
numbers of variables, i.e. indices 1 to $m$ mean auxiliary variables,
|
|
indices $m+1$ to $m+n$ mean structural ones.
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_transform_col| returns \verb|len'|, the number of
|
|
non-zero coefficients in the resultant column stored in the arrays
|
|
\verb|ind| and \verb|val|.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_prim\_rtest --- perform primal ratio test}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_prim_rtest(glp_prob *P, int len, const int ind[], const double val[],
|
|
int dir, double eps);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_prim_rtest| performs the primal ratio test using
|
|
an explicitly specified column of the simplex table.
|
|
|
|
The current basic solution associated with the LP problem object must
|
|
be primal feasible.
|
|
|
|
The explicitly specified column of the simplex table shows how the
|
|
basic variables $x_B$ depend on some non-basic variable $x$ (which is
|
|
not necessarily presented in the problem object):
|
|
$$
|
|
\begin{array}{l@{\ }c@{\ }r}
|
|
(x_B)_1&=\dots+&\xi_{1}x\\
|
|
(x_B)_2&=\dots+&\xi_{2}x\\
|
|
\multicolumn{3}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .}\\
|
|
(x_B)_m&=\dots+&\xi_{m}x\\
|
|
\end{array}
|
|
$$
|
|
|
|
The column is specifed on entry to the routine in sparse format.
|
|
Ordinal numbers of basic variables $(x_B)_i$ should be placed in
|
|
locations \verb|ind[1]|, \dots, \verb|ind[len]|, where ordinal number
|
|
1 to $m$ denote auxiliary variables, and ordinal numbers $m+1$ to $m+n$
|
|
denote structural variables. The corresponding non-zero coefficients
|
|
$\xi_i$ should be placed in locations
|
|
\verb|val[1]|, \dots, \verb|val[len]|. The arrays \verb|ind| and
|
|
\verb|val| are not changed by the routine.
|
|
|
|
The parameter \verb|dir| specifies direction in which the variable $x$
|
|
changes on entering the basis: $+1$ means increasing, $-1$ means
|
|
decreasing.
|
|
|
|
The parameter \verb|eps| is an absolute tolerance (small positive
|
|
number, say, $10^{-9}$) used by the routine to skip $\xi_i$'s whose
|
|
magnitude is less than \verb|eps|.
|
|
|
|
The routine determines which basic variable (among those specified in
|
|
\verb|ind[1]|, \dots, \verb|ind[len]|) reaches its (lower or upper)
|
|
bound first before any other basic variables do, and which, therefore,
|
|
should leave the basis in order to keep primal feasibility.
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_prim_rtest| returns the index, \verb|piv|, in the
|
|
arrays \verb|ind| and \verb|val| corresponding to the pivot element
|
|
chosen, $1\leq$ \verb|piv| $\leq$ \verb|len|. If the adjacent basic
|
|
solution is primal unbounded, and therefore the choice cannot be made,
|
|
the routine returns zero.
|
|
|
|
\para{Comments}
|
|
|
|
If the non-basic variable $x$ is presented in the LP problem object,
|
|
the input column can be computed with the routine
|
|
\verb|glp_eval_tab_col|; otherwise, it can be computed with the routine
|
|
\verb|glp_transform_col|.
|
|
|
|
\newpage
|
|
|
|
\subsection{glp\_dual\_rtest --- perform dual ratio test}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
int glp_dual_rtest(glp_prob *P, int len, const int ind[], const double val[],
|
|
int dir, double eps);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_dual_rtest| performs the dual ratio test using
|
|
an explicitly specified row of the simplex table.
|
|
|
|
The current basic solution associated with the LP problem object must
|
|
be dual feasible.
|
|
|
|
The explicitly specified row of the simplex table is a linear form
|
|
that shows how some basic variable $x$ (which is not necessarily
|
|
presented in the problem object) depends on non-basic variables $x_N$:
|
|
$$x=\xi_1(x_N)_1+\xi_2(x_N)_2+\dots+\xi_n(x_N)_n.$$
|
|
|
|
The row is specified on entry to the routine in sparse format. Ordinal
|
|
numbers of non-basic variables $(x_N)_j$ should be placed in locations
|
|
\verb|ind[1]|, \dots, \verb|ind[len]|, where ordinal numbers 1 to $m$
|
|
denote auxiliary variables, and ordinal numbers $m+1$ to $m+n$ denote
|
|
structural variables. The corresponding non-zero coefficients $\xi_j$
|
|
should be placed in locations \verb|val[1]|, \dots, \verb|val[len]|.
|
|
The arrays \verb|ind| and \verb|val| are not changed by the routine.
|
|
|
|
The parameter \verb|dir| specifies direction in which the variable $x$
|
|
changes on leaving the basis: $+1$ means that $x$ goes on its lower
|
|
bound, so its reduced cost (dual variable) is increasing (minimization)
|
|
or decreasing (maximization); $-1$ means that $x$ goes on its upper
|
|
bound, so its reduced cost is decreasing (minimization) or increasing
|
|
(maximization).
|
|
|
|
The parameter \verb|eps| is an absolute tolerance (small positive
|
|
number, say, $10^{-9}$) used by the routine to skip $\xi_j$'s whose
|
|
magnitude is less than \verb|eps|.
|
|
|
|
The routine determines which non-basic variable (among those specified
|
|
in \verb|ind[1]|, \dots,\linebreak \verb|ind[len]|) should enter the
|
|
basis in order to keep dual feasibility, because its reduced cost
|
|
reaches the (zero) bound first before this occurs for any other
|
|
non-basic variables.
|
|
|
|
\returns
|
|
|
|
The routine \verb|glp_dual_rtest| returns the index, \verb|piv|, in the
|
|
arrays \verb|ind| and \verb|val| corresponding to the pivot element
|
|
chosen, $1\leq$ \verb|piv| $\leq$ \verb|len|. If the adjacent basic
|
|
solution is dual unbounded, and therefore the choice cannot be made,
|
|
the routine returns zero.
|
|
|
|
\para{Comments}
|
|
|
|
If the basic variable $x$ is presented in the LP problem object, the
|
|
input row can be computed\linebreak with the routine
|
|
\verb|glp_eval_tab_row|; otherwise, it can be computed with the routine
|
|
\linebreak \verb|glp_transform_row|.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\newpage
|
|
|
|
\section{Post-optimal analysis routines}
|
|
|
|
\subsection{glp\_analyze\_bound --- analyze active bound of non-basic
|
|
variable}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_analyze_bound(glp_prob *P, int k, double *limit1, int *var1,
|
|
double *limit2, int *var2);
|
|
\end{verbatim}
|
|
|
|
\description
|
|
|
|
The routine \verb|glp_analyze_bound| analyzes the effect of varying the
|
|
active bound of specified non-basic variable.
|
|
|
|
The non-basic variable is specified by the parameter $k$, where
|
|
$1\leq k\leq m$ means auxiliary variable of corresponding row, and
|
|
$m+1\leq k\leq m+n$ means structural variable (column).
|
|
|
|
Note that the current basic solution must be optimal, and the basis
|
|
factorization must exist.
|
|
|
|
Results of the analysis have the following meaning.
|
|
|
|
\verb|value1| is the minimal value of the active bound, at which the
|
|
basis still remains primal feasible and thus optimal. \verb|-DBL_MAX|
|
|
means that the active bound has no lower limit.
|
|
|
|
\verb|var1| is the ordinal number of an auxiliary (1 to $m$) or
|
|
structural ($m+1$ to $m+n$) basic variable, which reaches its bound
|
|
first and thereby limits further decreasing the active bound being
|
|
analyzed. if \verb|value1| = \verb|-DBL_MAX|, \verb|var1| is set to 0.
|
|
|
|
\verb|value2| is the maximal value of the active bound, at which the
|
|
basis still remains primal feasible and thus optimal. \verb|+DBL_MAX|
|
|
means that the active bound has no upper limit.
|
|
|
|
\verb|var2| is the ordinal number of an auxiliary (1 to $m$) or
|
|
structural ($m+1$ to $m+n$) basic variable, which reaches its bound
|
|
first and thereby limits further increasing the active bound being
|
|
analyzed. if \verb|value2| = \verb|+DBL_MAX|, \verb|var2| is set to 0.
|
|
|
|
The parameters \verb|value1|, \verb|var1|, \verb|value2|, \verb|var2|
|
|
can be specified as \verb|NULL|, in which case corresponding information
|
|
is not stored.
|
|
|
|
\subsection{glp\_analyze\_coef --- analyze objective coefficient at
|
|
basic variable}
|
|
|
|
\synopsis
|
|
|
|
\begin{verbatim}
|
|
void glp_analyze_coef(glp_prob *P, int k,
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double *coef1, int *var1, double *value1,
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double *coef2, int *var2, double *value2);
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\end{verbatim}
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\description
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The routine \verb|glp_analyze_coef| analyzes the effect of varying the
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objective coefficient at specified basic variable.
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The basic variable is specified by the parameter $k$, where
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$1\leq k\leq m$ means auxiliary variable of corresponding row, and
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$m+1\leq k\leq m+n$ means structural variable (column).
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Note that the current basic solution must be optimal, and the basis
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factorization must exist.
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Results of the analysis have the following meaning.
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\verb|coef1| is the minimal value of the objective coefficient, at
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which the basis still remains dual feasible and thus optimal.
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\verb|-DBL_MAX| means that the objective coefficient has no lower
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limit.
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\verb|var1| is the ordinal number of an auxiliary (1 to $m$) or
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structural ($m+1$ to $m+n$) non-basic variable, whose reduced cost
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reaches its zero bound first and thereby limits further decreasing the
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objective coefficient being analyzed.
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If \verb|coef1| = \verb|-DBL_MAX|, \verb|var1| is set to 0.
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\verb|value1| is value of the basic variable being analyzed in an
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adjacent basis, which is defined as follows. Let the objective
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coefficient reach its minimal value (\verb|coef1|) and continue
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decreasing. Then the reduced cost of the limiting non-basic variable
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(\verb|var1|) becomes dual infeasible and the current basis becomes
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non-optimal that forces the limiting non-basic variable to enter the
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basis replacing there some basic variable that leaves the basis to keep
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primal feasibility. Should note that on determining the adjacent basis
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current bounds of the basic variable being analyzed are ignored as if
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it were free (unbounded) variable, so it cannot leave the basis. It may
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happen that no dual feasible adjacent basis exists, in which case
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\verb|value1| is set to \verb|-DBL_MAX| or \verb|+DBL_MAX|.
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\verb|coef2| is the maximal value of the objective coefficient, at
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which the basis still remains dual feasible and thus optimal.
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\verb|+DBL_MAX| means that the objective coefficient has no upper
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limit.
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\verb|var2| is the ordinal number of an auxiliary (1 to $m$) or
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structural ($m+1$ to $m+n$) non-basic variable, whose reduced cost
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reaches its zero bound first and thereby limits further increasing the
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objective coefficient being analyzed.
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If \verb|coef2| = \verb|+DBL_MAX|, \verb|var2| is set to 0.
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\verb|value2| is value of the basic variable being analyzed in an
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adjacent basis, which is defined exactly in the same way as
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\verb|value1| above with exception that now the objective coefficient
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is increasing.
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The parameters \verb|coef1|, \verb|var1|, \verb|value1|, \verb|coef2|,
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\verb|var2|, \verb|value2| can be specified as \verb|NULL|, in which
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case corresponding information is not stored.
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%* eof *%
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