You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
336 lines
12 KiB
336 lines
12 KiB
%* glpk01.tex *%
|
|
|
|
\chapter{Introduction}
|
|
|
|
GLPK (\underline{G}NU \underline{L}inear \underline{P}rogramming
|
|
\underline{K}it) is a set of routines written in the ANSI C programming
|
|
language and organized in the form of a callable library. It is
|
|
intended for solving linear programming (LP), mixed integer programming
|
|
(MIP), and other related problems.
|
|
|
|
\section{LP problem}
|
|
\label{seclp}
|
|
|
|
GLPK assumes the following formulation of the {\it linear programming
|
|
(LP)} problem:
|
|
|
|
\noindent
|
|
\hspace{.5in} minimize (or maximize)
|
|
$$z = c_1x_{m+1} + c_2x_{m+2} + \dots + c_nx_{m+n} + c_0 \eqno (1.1)$$
|
|
\hspace{.5in} subject to linear constraints
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}r@{\:}c@{\:}r@{\:}c@{\:}r}
|
|
x_1&=&a_{11}x_{m+1}&+&a_{12}x_{m+2}&+ \dots +&a_{1n}x_{m+n} \\
|
|
x_2&=&a_{21}x_{m+1}&+&a_{22}x_{m+2}&+ \dots +&a_{2n}x_{m+n} \\
|
|
\multicolumn{7}{c}
|
|
{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .} \\
|
|
x_m&=&a_{m1}x_{m+1}&+&a_{m2}x_{m+2}&+ \dots +&a_{mn}x_{m+n} \\
|
|
\end{array} \eqno (1.2)
|
|
$$
|
|
\hspace{.5in} and bounds of variables
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}c@{\:}c@{\:}l}
|
|
l_1&\leq&x_1&\leq&u_1 \\
|
|
l_2&\leq&x_2&\leq&u_2 \\
|
|
\multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .}\\
|
|
l_{m+n}&\leq&x_{m+n}&\leq&u_{m+n} \\
|
|
\end{array} \eqno (1.3)
|
|
$$
|
|
where: $x_1, x_2, \dots, x_m$ are auxiliary variables;
|
|
$x_{m+1}, x_{m+2}, \dots, x_{m+n}$ are structural variables;
|
|
$z$ is the objective function;
|
|
$c_1, c_2, \dots, c_n$ are objective coefficients;
|
|
$c_0$ is the constant term (``shift'') of the objective function;
|
|
$a_{11}, a_{12}, \dots, a_{mn}$ are constraint coefficients;
|
|
$l_1, l_2, \dots, l_{m+n}$ are lower bounds of variables;
|
|
$u_1, u_2, \dots, u_{m+n}$ are upper bounds of variables.
|
|
|
|
Auxiliary variables are also called {\it rows}, because they correspond
|
|
to rows of the constraint matrix (i.e. a matrix built of the constraint
|
|
coefficients). Similarly, structural variables are also called
|
|
{\it columns}, because they correspond to columns of the constraint
|
|
matrix.
|
|
|
|
Bounds of variables can be finite as well as infinite. Besides, lower
|
|
and upper bounds can be equal to each other. Thus, the following types
|
|
of variables are possible:
|
|
|
|
\begin{center}
|
|
\begin{tabular}{r@{}c@{}ll}
|
|
\multicolumn{3}{c}{Bounds of variable} & Type of variable \\
|
|
\hline
|
|
$-\infty <$ &$\ x_k\ $& $< +\infty$ & Free (unbounded) variable \\
|
|
$l_k \leq$ &$\ x_k\ $& $< +\infty$ & Variable with lower bound \\
|
|
$-\infty <$ &$\ x_k\ $& $\leq u_k$ & Variable with upper bound \\
|
|
$l_k \leq$ &$\ x_k\ $& $\leq u_k$ & Double-bounded variable \\
|
|
$l_k =$ &$\ x_k\ $& $= u_k$ & Fixed variable \\
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
\noindent
|
|
Note that the types of variables shown above are applicable to
|
|
structural as well as to auxiliary variables.
|
|
|
|
To solve the LP problem (1.1)---(1.3) is to find such values of all
|
|
structural and auxiliary variables, which:
|
|
|
|
%\vspace*{-10pt}
|
|
|
|
%\begin{itemize}\setlength{\itemsep}{0pt}
|
|
\Item{---}satisfy to all the linear constraints (1.2), and
|
|
|
|
\Item{---}are within their bounds (1.3), and
|
|
|
|
\Item{---}provide smallest (in case of minimization) or largest (in
|
|
case of maximization) value of the objective function (1.1).
|
|
%\end{itemize}
|
|
|
|
\section{MIP problem}
|
|
|
|
{\it Mixed integer linear programming (MIP)} problem is an LP problem
|
|
in which some variables are additionally required to be integer.
|
|
|
|
GLPK assumes that MIP problem has the same formulation as ordinary
|
|
(pure) LP problem (1.1)---(1.3), i.e. includes auxiliary and structural
|
|
variables, which may have lower and/or upper bounds. However, in case
|
|
of MIP problem some variables may be required to be integer. This
|
|
additional constraint means that a value of each {\it integer variable}
|
|
must be only integer number. (Should note that GLPK allows only
|
|
structural variables to be of integer kind.)
|
|
|
|
\section{Using the package}
|
|
|
|
\subsection{Brief example}
|
|
|
|
In order to understand what GLPK is from the user's standpoint,
|
|
consider the following simple LP problem:
|
|
|
|
\noindent
|
|
\hspace{.5in} maximize
|
|
$$z = 10 x_1 + 6 x_2 + 4 x_3$$
|
|
\hspace{.5in} subject to
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}r@{\:}c@{\:}r@{\:}c@{\:}r}
|
|
x_1 &+&x_2 &+&x_3 &\leq 100 \\
|
|
10 x_1 &+& 4 x_2 & +&5 x_3 & \leq 600 \\
|
|
2 x_1 &+& 2 x_2 & +& 6 x_3 & \leq 300 \\
|
|
\end{array}
|
|
$$
|
|
\hspace{.5in} where all variables are non-negative
|
|
$$x_1 \geq 0, \ x_2 \geq 0, \ x_3 \geq 0$$
|
|
|
|
At first, this LP problem should be transformed to the standard form
|
|
(1.1)---(1.3). This can be easily done by introducing auxiliary
|
|
variables, by one for each original inequality constraint. Thus, the
|
|
problem can be reformulated as follows:
|
|
|
|
\noindent
|
|
\hspace{.5in} maximize
|
|
$$z = 10 x_1 + 6 x_2 + 4 x_3$$
|
|
\hspace{.5in} subject to
|
|
$$
|
|
\begin{array}{r@{\:}c@{\:}r@{\:}c@{\:}r@{\:}c@{\:}r}
|
|
p& = &x_1 &+&x_2 &+&x_3 \\
|
|
q& = &10 x_1 &+& 4 x_2 &+& 5 x_3 \\
|
|
r& = &2 x_1 &+& 2 x_2 &+& 6 x_3 \\
|
|
\end{array}
|
|
$$
|
|
\hspace{.5in} and bounds of variables
|
|
$$
|
|
\begin{array}{ccc}
|
|
\nonumber -\infty < p \leq 100 && 0 \leq x_1 < +\infty \\
|
|
\nonumber -\infty < q \leq 600 && 0 \leq x_2 < +\infty \\
|
|
\nonumber -\infty < r \leq 300 && 0 \leq x_3 < +\infty \\
|
|
\end{array}
|
|
$$
|
|
where $p, q, r$ are auxiliary variables (rows), and $x_1, x_2, x_3$ are
|
|
structural variables (columns).
|
|
|
|
The example C program shown below uses GLPK API routines in order to
|
|
solve this LP problem.\footnote{If you just need to solve LP or MIP
|
|
instance, you may write it in MPS or CPLEX LP format and then use the
|
|
GLPK stand-alone solver to obtain a solution. This is much less
|
|
time-consuming than programming in C with GLPK API routines.}
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
/* sample.c */
|
|
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
#include <glpk.h>
|
|
|
|
int main(void)
|
|
{ glp_prob *lp;
|
|
int ia[1+1000], ja[1+1000];
|
|
double ar[1+1000], z, x1, x2, x3;
|
|
s1: lp = glp_create_prob();
|
|
s2: glp_set_prob_name(lp, "sample");
|
|
s3: glp_set_obj_dir(lp, GLP_MAX);
|
|
s4: glp_add_rows(lp, 3);
|
|
s5: glp_set_row_name(lp, 1, "p");
|
|
s6: glp_set_row_bnds(lp, 1, GLP_UP, 0.0, 100.0);
|
|
s7: glp_set_row_name(lp, 2, "q");
|
|
s8: glp_set_row_bnds(lp, 2, GLP_UP, 0.0, 600.0);
|
|
s9: glp_set_row_name(lp, 3, "r");
|
|
s10: glp_set_row_bnds(lp, 3, GLP_UP, 0.0, 300.0);
|
|
s11: glp_add_cols(lp, 3);
|
|
s12: glp_set_col_name(lp, 1, "x1");
|
|
s13: glp_set_col_bnds(lp, 1, GLP_LO, 0.0, 0.0);
|
|
s14: glp_set_obj_coef(lp, 1, 10.0);
|
|
s15: glp_set_col_name(lp, 2, "x2");
|
|
s16: glp_set_col_bnds(lp, 2, GLP_LO, 0.0, 0.0);
|
|
s17: glp_set_obj_coef(lp, 2, 6.0);
|
|
s18: glp_set_col_name(lp, 3, "x3");
|
|
s19: glp_set_col_bnds(lp, 3, GLP_LO, 0.0, 0.0);
|
|
s20: glp_set_obj_coef(lp, 3, 4.0);
|
|
s21: ia[1] = 1, ja[1] = 1, ar[1] = 1.0; /* a[1,1] = 1 */
|
|
s22: ia[2] = 1, ja[2] = 2, ar[2] = 1.0; /* a[1,2] = 1 */
|
|
s23: ia[3] = 1, ja[3] = 3, ar[3] = 1.0; /* a[1,3] = 1 */
|
|
s24: ia[4] = 2, ja[4] = 1, ar[4] = 10.0; /* a[2,1] = 10 */
|
|
s25: ia[5] = 3, ja[5] = 1, ar[5] = 2.0; /* a[3,1] = 2 */
|
|
s26: ia[6] = 2, ja[6] = 2, ar[6] = 4.0; /* a[2,2] = 4 */
|
|
s27: ia[7] = 3, ja[7] = 2, ar[7] = 2.0; /* a[3,2] = 2 */
|
|
s28: ia[8] = 2, ja[8] = 3, ar[8] = 5.0; /* a[2,3] = 5 */
|
|
s29: ia[9] = 3, ja[9] = 3, ar[9] = 6.0; /* a[3,3] = 6 */
|
|
s30: glp_load_matrix(lp, 9, ia, ja, ar);
|
|
s31: glp_simplex(lp, NULL);
|
|
s32: z = glp_get_obj_val(lp);
|
|
s33: x1 = glp_get_col_prim(lp, 1);
|
|
s34: x2 = glp_get_col_prim(lp, 2);
|
|
s35: x3 = glp_get_col_prim(lp, 3);
|
|
s36: printf("\nz = %g; x1 = %g; x2 = %g; x3 = %g\n",
|
|
z, x1, x2, x3);
|
|
s37: glp_delete_prob(lp);
|
|
return 0;
|
|
}
|
|
|
|
/* eof */
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
The statement \verb|s1| creates a problem object. Being created the
|
|
object is initially empty. The statement \verb|s2| assigns a symbolic
|
|
name to the problem object.
|
|
|
|
The statement \verb|s3| calls the routine \verb|glp_set_obj_dir| in
|
|
order to set the optimization direction flag, where \verb|GLP_MAX|
|
|
means maximization.
|
|
|
|
The statement \verb|s4| adds three rows to the problem object.
|
|
|
|
The statement \verb|s5| assigns the symbolic name `\verb|p|' to the
|
|
first row, and the statement \verb|s6| sets the type and bounds of the
|
|
first row, where \verb|GLP_UP| means that the row has an upper bound.
|
|
The statements \verb|s7|, \verb|s8|, \verb|s9|, \verb|s10| are used in
|
|
the same way in order to assign the symbolic names `\verb|q|' and
|
|
`\verb|r|' to the second and third rows and set their types and bounds.
|
|
|
|
The statement \verb|s11| adds three columns to the problem object.
|
|
|
|
The statement \verb|s12| assigns the symbolic name `\verb|x1|' to the
|
|
first column, the statement \verb|s13| sets the type and bounds of the
|
|
first column, where \verb|GLP_LO| means that the column has an lower
|
|
bound, and the statement \verb|s14| sets the objective coefficient for
|
|
the first column. The statements \verb|s15|---\verb|s20| are used in
|
|
the same way in order to assign the symbolic names `\verb|x2|' and
|
|
`\verb|x3|' to the second and third columns and set their types,
|
|
bounds, and objective coefficients.
|
|
|
|
The statements \verb|s21|---\verb|s29| prepare non-zero elements of the
|
|
constraint matrix (i.e. constraint coefficients). Row indices of each
|
|
element are stored in the array \verb|ia|, column indices are stored in
|
|
the array \verb|ja|, and numerical values of corresponding elements are
|
|
stored in the array \verb|ar|. Then the statement \verb|s30| calls
|
|
the routine \verb|glp_load_matrix|, which loads information from these
|
|
three arrays into the problem object.
|
|
|
|
Now all data have been entered into the problem object, and therefore
|
|
the statement \verb|s31| calls the routine \verb|glp_simplex|, which is
|
|
a driver to the simplex method, in order to solve the LP problem. This
|
|
routine finds an optimal solution and stores all relevant information
|
|
back into the problem object.
|
|
|
|
The statement \verb|s32| obtains a computed value of the objective
|
|
function, and the statements \verb|s33|---\verb|s35| obtain computed
|
|
values of structural variables (columns), which correspond to the
|
|
optimal basic solution found by the solver.
|
|
|
|
The statement \verb|s36| writes the optimal solution to the standard
|
|
output. The printout may look like follows:
|
|
|
|
\newpage
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
* 0: objval = 0.000000000e+00 infeas = 0.000000000e+00 (0)
|
|
* 2: objval = 7.333333333e+02 infeas = 0.000000000e+00 (0)
|
|
OPTIMAL SOLUTION FOUND
|
|
|
|
z = 733.333; x1 = 33.3333; x2 = 66.6667; x3 = 0
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
Finally, the statement \verb|s37| calls the routine
|
|
\verb|glp_delete_prob|, which frees all the memory allocated to the
|
|
problem object.
|
|
|
|
\subsection{Compiling}
|
|
|
|
The GLPK package has the only header file \verb|glpk.h|, which should
|
|
be available on compiling a C (or C++) program using GLPK API routines.
|
|
|
|
If the header file is installed in the default location
|
|
\verb|/usr/local/include|, the following typical command may be used to
|
|
compile, say, the example C program described above with the GNU C
|
|
compiler:
|
|
|
|
\begin{verbatim}
|
|
$ gcc -c sample.c
|
|
\end{verbatim}
|
|
|
|
If \verb|glpk.h| is not in the default location, the corresponding
|
|
directory containing it should be made known to the C compiler through
|
|
\verb|-I| option, for example:
|
|
|
|
\begin{verbatim}
|
|
$ gcc -I/foo/bar/glpk-4.15/include -c sample.c
|
|
\end{verbatim}
|
|
|
|
In any case the compilation results in an object file \verb|sample.o|.
|
|
|
|
\subsection{Linking}
|
|
|
|
The GLPK library is a single file \verb|libglpk.a|. (On systems which
|
|
support shared libraries there may be also a shared version of the
|
|
library \verb|libglpk.so|.)
|
|
|
|
If the library is installed in the default
|
|
location \verb|/usr/local/lib|, the following typical command may be
|
|
used to link, say, the example C program described above against with
|
|
the library:
|
|
|
|
\begin{verbatim}
|
|
$ gcc sample.o -lglpk -lm
|
|
\end{verbatim}
|
|
|
|
If the GLPK library is not in the default location, the corresponding
|
|
directory containing it should be made known to the linker through
|
|
\verb|-L| option, for example:
|
|
|
|
\begin{verbatim}
|
|
$ gcc -L/foo/bar/glpk-4.15 sample.o -lglpk -lm
|
|
\end{verbatim}
|
|
|
|
Depending on configuration of the package linking against with the GLPK
|
|
library may require optional libraries, in which case these libraries
|
|
should be also made known to the linker, for example:
|
|
|
|
\begin{verbatim}
|
|
$ gcc sample.o -lglpk -lgmp -lm
|
|
\end{verbatim}
|
|
|
|
For more details about configuration options of the GLPK package see
|
|
Appendix \ref{install}, page \pageref{install}.
|
|
|
|
%* eof *%
|