(octave.info)Linear Programming
25.1 Linear Programming
=======================
Octave can solve Linear Programming problems using the ‘glpk’ function.
That is, Octave can solve
min C'*x
subject to the linear constraints A*x = b where x ≥ 0.
The ‘glpk’ function also supports variations of this problem.
-- : [XOPT, FMIN, ERRNUM, EXTRA] = glpk (C, A, B, LB, UB, CTYPE,
VARTYPE, SENSE, PARAM)
Solve a linear program using the GNU GLPK library.
Given three arguments, ‘glpk’ solves the following standard LP:
min C'*x
subject to
A*x = b
x >= 0
but may also solve problems of the form
[ min | max ] C'*x
subject to
A*x [ "=" | "<=" | ">=" ] b
x >= LB
x <= UB
Input arguments:
C
A column array containing the objective function coefficients.
A
A matrix containing the constraints coefficients.
B
A column array containing the right-hand side value for each
constraint in the constraint matrix.
LB
An array containing the lower bound on each of the variables.
If LB is not supplied, the default lower bound for the
variables is zero.
UB
An array containing the upper bound on each of the variables.
If UB is not supplied, the default upper bound is assumed to
be infinite.
CTYPE
An array of characters containing the sense of each constraint
in the constraint matrix. Each element of the array may be
one of the following values
"F"
A free (unbounded) constraint (the constraint is
ignored).
"U"
An inequality constraint with an upper bound (‘A(i,:)*x
<= b(i)’).
"S"
An equality constraint (‘A(i,:)*x = b(i)’).
"L"
An inequality with a lower bound (‘A(i,:)*x >= b(i)’).
"D"
An inequality constraint with both upper and lower bounds
(‘A(i,:)*x >= -b(i)’) _and_ (‘A(i,:)*x <= b(i)’).
VARTYPE
A column array containing the types of the variables.
"C"
A continuous variable.
"I"
An integer variable.
SENSE
If SENSE is 1, the problem is a minimization. If SENSE is -1,
the problem is a maximization. The default value is 1.
PARAM
A structure containing the following parameters used to define
the behavior of solver. Missing elements in the structure
take on default values, so you only need to set the elements
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).
‘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
specified, in which case the routine chooses the scaling
options automatically.
‘dual (default: 1)’
Simplex method option:
1 (‘GLP_PRIMAL’)
Use two-phase primal simplex.
2 (‘GLP_DUALP’)
Use two-phase dual simplex, and if it fails, switch
to the primal simplex.
3 (‘GLP_DUAL’)
Use two-phase dual simplex.
‘price (default: 34)’
Pricing option (for both primal and dual simplex):
17 (‘GLP_PT_STD’)
Textbook pricing.
34 (‘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 built-in
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: 34)’
Ratio test technique:
17 (‘GLP_RT_STD’)
Standard ("textbook").
34 (‘GLP_RT_HAR’)
Harris’ two-pass 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 (default: 0)’
If this parameter is nonzero, save a copy of the problem
in CPLEX LP format to the file ‘"outpb.lp"’. There is
currently no way to change the name of the output file.
Real parameters:
‘tolbnd (default: 1e-7)’
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: 1e-7)’
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: 1e-10)’
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: -DBL_MAX)’
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: +DBL_MAX)’
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: 1e-5)’
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: 1e-7)’
Relative tolerance used to check if the value of the
objective function is not better than in the best known
integer feasible 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.
ERRNUM
Error code.
0
No error.
1 (‘GLP_EBADB’)
Invalid basis.
2 (‘GLP_ESING’)
Singular matrix.
3 (‘GLP_ECOND’)
Ill-conditioned matrix.
4 (‘GLP_EBOUND’)
Invalid bounds.
5 (‘GLP_EFAIL’)
Solver failed.
6 (‘GLP_EOBJLL’)
Objective function lower limit reached.
7 (‘GLP_EOBJUL’)
Objective function upper limit reached.
8 (‘GLP_EITLIM’)
Iterations limit exhausted.
9 (‘GLP_ETMLIM’)
Time limit exhausted.
10 (‘GLP_ENOPFS’)
No primal feasible solution.
11 (‘GLP_ENODFS’)
No dual feasible solution.
12 (‘GLP_EROOT’)
Root LP optimum not provided.
13 (‘GLP_ESTOP’)
Search terminated by application.
14 (‘GLP_EMIPGAP’)
Relative MIP gap tolerance reached.
15 (‘GLP_ENOFEAS’)
No primal/dual feasible solution.
16 (‘GLP_ENOCVG’)
No convergence.
17 (‘GLP_EINSTAB’)
Numerical instability.
18 (‘GLP_EDATA’)
Invalid data.
19 (‘GLP_ERANGE’)
Result out of range.
EXTRA
A data structure containing the following fields:
‘lambda’
Dual variables.
‘redcosts’
Reduced Costs.
‘time’
Time (in seconds) used for solving LP/MIP problem.
‘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.
Example:
c = [10, 6, 4]';
A = [ 1, 1, 1;
10, 4, 5;
2, 2, 6];
b = [100, 600, 300]';
lb = [0, 0, 0]';
ub = [];
ctype = "UUU";
vartype = "CCC";
s = -1;
param.msglev = 1;
param.itlim = 100;
[xmin, fmin, status, extra] = ...
glpk (c, A, b, lb, ub, ctype, vartype, s, param);
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