NAME
PDL::Opt::GLPK  PDL interface to the GNU Linear Programming Kit
SYNOPSIS
use PDL;
use PDL::Opt::GLPK;
glpk($c, $a, $b $lb, $ub, $ctype, $vtype, GLP_MAX,
$xopt = null, $fopt = null, $status = null,
$lambda = null, $redcosts = null, \%param);
DESCRIPTION
This module provides an interface to GLPK, the GNU Linear Programming Kit. The interface was ported from Octave and mimics its GLPK interface.
FUNCTIONS
glpk
Signature (c(m); a(m, n); b(n); lb(m); ub(m); ctype(n); vtype(m);
int sense; [o]xopt(m); [o]fopt(); [o]status(); [o]lambda(n);
[o]redcosts(n); SV *param)
Solve a linear program using the GNU GLPK library.
The LP can be of the form
[ min  max ] C'*x
subject to
A*x [ "="  "<="  ">=" ] b
x >= LB
x <= UB
Arguments
Input values:

c
A ndarray containing the objective function coefficients.

a
A ndarray containing the constraints coefficients.
a
must be 2d, though broadcasting over higher dimensions is possible for other arguments.The coefficients may be given as a PDL::CCS::Nd sparse matrix as well.

b
A ndarray containing the righthand side value for each constraint in the constraint matrix.

lb
A ndarray containing the lower bound on each of the variables.

ub
A ndarray containing the upper bound on each of the variables.

ctype
A ndarray containing the sense of each constraint in the constraint matrix. Each element of the array may be one of the following values

1 (GLP_FR)
A free (unbounded) constraint (the constraint is ignored).

2 (GLP_LO)
An inequality constraint with a lower bound.

3 (GLP_UP)
An inequality constraint with an upper bound.

4 (GLP_DB)
An inequality constraint with equal lower and upper bounds.

5 (GLP_FX)
An equality constraint.


vartype
A ndarray containing the types of the variables.

1 (GLP_CV)
A continuous variable.

2 (GLP_IV)
An integer variable.

3 (GLP_BV)
A binary variable. Lower and upper bound are ignored and set to zero resp. one.

 sense

1 (GLP_MIN)
Minimize the objective function.

2 (GLP_MAX)
Maximize the objective function.


param (optional)
A hash reference with any of following keys used to define the behavior of solver. Missing keys take on default values, so you only need to set the values that you wish to change from the default.
Integer parameters:

msglev (default: 1)
Level of messages output by solver routines:

0 (GLP_MSG_OFF)
No output.

1 (GLP_MSG_ERR)
Error and warning messages only.

2 (GLP_MSG_ON)
Normal output.

3 (GLP_MSG_ALL)
Full output (includes informational messages).

4 (GLP_MSG_DBG)
Debug output.


scale (default: 16)
Scaling option. The values can be combined with the bitwise OR operator and may be the following:

1 (GLP_SF_GM)
Geometric mean scaling.

16 (GLP_SF_EQ)
Equilibration scaling.

32 (GLP_SF_2N)
Round scale factors to power of two.

64 (GLP_SF_SKIP)
Skip if problem is well scaled.
Alternatively, a value of 128 (GLP_SF_AUTO) may be also spec ified, in which case the routine chooses the scaling options automatically.


dual (default: 1)
Simplex method option:

1 (GLP_PRIMAL)
Use twophase primal simplex.

2 (GLP_DUALP)
Use twophase dual simplex, and if it fails, switch to the primal simplex.

3 (GLP_DUAL)
Use twophase dual simplex.


price (default: 0x22)
Pricing option (for both primal and dual simplex):

0x11 (GLP_PT_STD)
Textbook pricing.

0x22 (GLP_PT_PSE)
Steepest edge pricing.


itlim (default: intmax)
Simplex iterations limit. It is decreased by one each time when one simplex iteration has been performed, and reaching zero value signals the solver to stop the search.

outfrq (default: 200)
Output frequency, in iterations. This parameter specifies how frequently the solver sends information about the solution to the standard output.

branch (default: 4)
Branching technique option (for MIP only):

1 (GLP_BR_FFV)
First fractional variable.

2 (GLP_BR_LFV)
Last fractional variable.

3 (GLP_BR_MFV)
Most fractional variable.

4 (GLP_BR_DTH)
Heuristic by Driebeck and Tomlin.

5 (GLP_BR_PCH)
Hybrid pseudocost heuristic.


btrack (default: 4)
Backtracking technique option (for MIP only):

1 (GLP_BT_DFS)
Depth first search.

2 (GLP_BT_BFS)
Breadth first search.

3 (GLP_BT_BLB)
Best local bound.

4 (GLP_BT_BPH)
Best projection heuristic.


presol (default: 1)
If this flag is set, the simplex solver uses the builtin LP presolver. Otherwise the LP presolver is not used.

lpsolver (default: 1)
Select which solver to use. If the problem is a MIP problem this flag will be ignored.

1 Revised simplex method.

2 Interior point method.


rtest (default: 0x22)
Ratio test technique:

0x11 (GLP_RT_STD)
Standard (“textbook”).

0x22 (GLP_RT_HAR)
Harris’ twopass ratio test.

0x33 (GLP_RT_FLIP)
longstep (flipflop) ratio test


tmlim (default: intmax)
Searching time limit, in milliseconds.

outdly (default: 0)
Output delay, in seconds. This parameter specifies how long the solver should delay sending information about the solution to the standard output.

save_pb (default: 0)
If this parameter is nonzero, save a copy of the problem in CPLEX LP format to a file as specified by the parameter
save_fn
. 
save_fn (default:
"outpb.%d.lp"
)This is a format for the file name(s) the problem will be written to if
save_pb
is true. The pattern shall contain a placeholder like%d
that will be replaced with a sequence number. This is required for broadcasting to prevent the file being overwritten.
Real parameters:

tolbnd (default: 1e7)
Relative tolerance used to check if the current basic solution is primal feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

toldj (default: 1e7)
Absolute tolerance used to check if the current basic solution is dual feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

tolpiv (default: 1e10)
Relative tolerance used to choose eligible pivotal elements of the simplex table. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

objll (default: INFINITY)
Lower limit of the objective function. If the objective function reaches this limit and continues decreasing, the solver stops the search. This parameter is used in the dual simplex method only.

objul (default: INFINITY)
Upper limit of the objective function. If the objective function reaches this limit and continues increasing, the solver stops the search. This parameter is used in the dual simplex only.

tolint (default: 1e5)
Relative tolerance used to check if the current basic solution is integer feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

tolobj (default: 1e7)
Relative tolerance used to check if the value of the objective function is not better than in the best known integer feasi ble solution. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

Output values:

xopt
The optimizer (the value of the decision variables at the optimum).

fopt
The optimum value of the objective function.

lambda (optional)
Dual variables.

redcosts (optional)
Reduced Costs.

status Status of the optimization.

1 (GLP_UNDEF)
Solution status is undefined.

2 (GLP_FEAS)
Solution is feasible.

3 (GLP_INFEAS)
Solution is infeasible.

4 (GLP_NOFEAS)
Problem has no feasible solution.

5 (GLP_OPT)
Solution is optimal.

6 (GLP_UNBND)
Problem has no unbounded solution.

EXAMPLES
A standard case
A modified example from the GLPK documentation, with an extra variable to avoid a square matrix:
Maximize
obj: + 10 x_1 + 6 x_2 + 4 x_3  x_4
Subject To
r_1: + x_1 + x_2 + x_3 + x_4 <= 100
r_2: + 10 x_1 + 4 x_2 + 5 x_3 + x_4 <= 600
r_3: + 2 x_1 + 2 x_2 + 6 x_3 + x_4 <= 300
The solution is straightforward:
use PDL;
use PDL::Opt::GLPK;
$a = pdl([[1, 1, 1, 1], [10, 4, 5, 1], [2, 2, 6, 1]]);
$b = pdl([100, 600, 300]);
$c = pdl([10, 6, 4, 1]);
$lb = zeroes(4);
$ub = inf(4);
$ctype = pdl([GLP_UP, GLP_UP, GLP_UP]);
$vtype = pdl([GLP_CV, GLP_CV, GLP_CV, GLP_CV]);
glpk($c, $a, $b $lb, $ub, $ctype, $vtype, GLPX_MAX,
$xopt = null, $fopt = null, $status = null);
# $xopt:
# [
# [ 33.333333 66.666667 0 0]
# ]
# $fopt:
# [ 733.33333]
Broadcasting
Multiple problems on the same coefficients may be solved simultaneously. Some care must be taken when all combinations of multiple arguments are requested.
The base problem:
Maximize
obj: + y_1 + y_2 + y_3 + y_4
Subject To
r_1:  y_2 + y_1 >= 1
r_2:  y_3 + y_2 >= 1
r_3:  y_4 + y_3 >= 1
Bounds
0 <= y_1 <= 4
0 <= y_2 <= 4
0 <= y_3 <= 4
0 <= y_4 <= 4
Generals
y_1
y_2
y_3
y_4
Looking for the objective function’s minimum and maximum with both lower and upper bound constraints:
use PDL;
use PDL::Opt::GLPK;
my $a = pdl(
[[1, 1, 0, 0],
[0, 1, 1, 0],
[0, 0, 1, 1]]);
my $b = pdl([1, 1, 1]);
my $c = pdl([1, 1, 1, 1]);
my $lb = pdl([0, 0, 0, 0]);
my $ub = pdl([4, 4, 4, 4]);
# dims: 3, 2  loop over lower and upper bounds
my $ctype = pdl([[GLP_LO, GLP_LO, GLP_LO],[GLP_UP, GLP_UP, GLP_UP]]);
my $vtype = (GLP_IV * ones(4));
# dims: 1, 2  extra loop over min and max
my $sense = pdl [[GLPX_MAX], [GLPX_MIN]];
my $xopt = null;
my $fopt = null;
my $status = null;
glpk($c, $a, $b, $lb, $ub, $ctype, $vtype, $sense, $xopt, $fopt, $status);
# $xopt:
# [
# [
# [4 3 2 1]
# [4 4 4 4]
# ]
# [
# [3 2 1 0]
# [0 0 0 0]
# ]
# ]
#
# $fopt:
# [
# [10 16]
# [ 6 0]
# ]
Specifying parameters
The params
hash ref is always the last argument. It is permitted to
leave out the other optional output arguments, as in
glpk($c, $a, $b, $lb, $up, $ctpye, $vtype, $sense,
$xopt, $fopt, $status, {save_pb => 1});
ERRORS
In case the solver reports an error, it will raise an PDL error.
AUTHOR
Jörg Sommrey
COPYRIGHT AND LICENSE
Copyright 2024 Jörg Sommrey
This library is free software; you may redistribute it and/or modify it under the terms of the GNU GENERAL PUBLIC LICENSE Version 3. See COPYING.
SEE ALSO
The Octave Manual: https://docs.octave.org/latest/LinearProgramming.html
The GNU Linear Programming Kit: https://www.gnu.org/software/glpk/
The GLPK reference manual is included in the GLPK distribution.