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<hr>
<h3 class="section" id="Linear-Programming-1"><span>25.1 Linear Programming<a class="copiable-link" href="#Linear-Programming-1"> ¶</a></span></h3>
<p>Octave can solve Linear Programming problems using the <code class="code">glpk</code>
function. That is, Octave can solve
</p>
<div class="example">
<pre class="example-preformatted">min C'*x
</pre></div>
<p>subject to the linear constraints
<em class="math">A*x = b</em> where <em class="math">x ≥ 0</em>.
</p>
<p>The <code class="code">glpk</code> function also supports variations of this problem.
</p>
<a class="anchor" id="XREFglpk"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-glpk"><span><code class="def-type">[<var class="var">xopt</var>, <var class="var">fmin</var>, <var class="var">errnum</var>, <var class="var">extra</var>] =</code> <strong class="def-name">glpk</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">A</var>, <var class="var">b</var>, <var class="var">lb</var>, <var class="var">ub</var>, <var class="var">ctype</var>, <var class="var">vartype</var>, <var class="var">sense</var>, <var class="var">param</var>)</code><a class="copiable-link" href="#index-glpk"> ¶</a></span></dt>
<dd><p>Solve a linear program using the GNU <small class="sc">GLPK</small> library.
</p>
<p>Given three arguments, <code class="code">glpk</code> solves the following standard LP:
</p>
<div class="example">
<pre class="example-preformatted">min C'*x
</pre></div>
<p>subject to
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">A*x = b
x >= 0
</pre></div></div>
<p>but may also solve problems of the form
</p>
<div class="example">
<pre class="example-preformatted">[ min | max ] C'*x
</pre></div>
<p>subject to
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">A*x [ "=" | "<=" | ">=" ] b
x >= LB
x <= UB
</pre></div></div>
<p>Input arguments:
</p>
<dl class="table">
<dt><var class="var">c</var></dt>
<dd><p>A column array containing the objective function coefficients.
</p>
</dd>
<dt><var class="var">A</var></dt>
<dd><p>A matrix containing the constraints coefficients.
</p>
</dd>
<dt><var class="var">b</var></dt>
<dd><p>A column array containing the right-hand side value for each constraint in
the constraint matrix.
</p>
</dd>
<dt><var class="var">lb</var></dt>
<dd><p>An array containing the lower bound on each of the variables. If <var class="var">lb</var>
is not supplied, the default lower bound for the variables is zero.
</p>
</dd>
<dt><var class="var">ub</var></dt>
<dd><p>An array containing the upper bound on each of the variables. If <var class="var">ub</var>
is not supplied, the default upper bound is assumed to be infinite.
</p>
</dd>
<dt><var class="var">ctype</var></dt>
<dd><p>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
</p>
<dl class="table">
<dt><code class="code">"F"</code></dt>
<dd><p>A free (unbounded) constraint (the constraint is ignored).
</p>
</dd>
<dt><code class="code">"U"</code></dt>
<dd><p>An inequality constraint with an upper bound (<code class="code">A(i,:)*x <= b(i)</code>).
</p>
</dd>
<dt><code class="code">"S"</code></dt>
<dd><p>An equality constraint (<code class="code">A(i,:)*x = b(i)</code>).
</p>
</dd>
<dt><code class="code">"L"</code></dt>
<dd><p>An inequality with a lower bound (<code class="code">A(i,:)*x >= b(i)</code>).
</p>
</dd>
<dt><code class="code">"D"</code></dt>
<dd><p>An inequality constraint with both upper and lower bounds
(<code class="code">A(i,:)*x >= -b(i)</code>) <em class="emph">and</em> (<code class="code">A(i,:)*x <= b(i)</code>).
</p></dd>
</dl>
</dd>
<dt><var class="var">vartype</var></dt>
<dd><p>A column array containing the types of the variables.
</p>
<dl class="table">
<dt><code class="code">"C"</code></dt>
<dd><p>A continuous variable.
</p>
</dd>
<dt><code class="code">"I"</code></dt>
<dd><p>An integer variable.
</p></dd>
</dl>
</dd>
<dt><var class="var">sense</var></dt>
<dd><p>If <var class="var">sense</var> is 1, the problem is a minimization. If <var class="var">sense</var> is -1,
the problem is a maximization. The default value is 1.
</p>
</dd>
<dt><var class="var">param</var></dt>
<dd><p>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.
</p>
<p>Integer parameters:
</p>
<dl class="table">
<dt><code class="code">msglev (default: 1)</code></dt>
<dd><p>Level of messages output by solver routines:
</p>
<dl class="table">
<dt>0 (<code class="code">GLP_MSG_OFF</code><!-- /@w -->)</dt>
<dd><p>No output.
</p>
</dd>
<dt>1 (<code class="code">GLP_MSG_ERR</code><!-- /@w -->)</dt>
<dd><p>Error and warning messages only.
</p>
</dd>
<dt>2 (<code class="code">GLP_MSG_ON</code><!-- /@w -->)</dt>
<dd><p>Normal output.
</p>
</dd>
<dt>3 (<code class="code">GLP_MSG_ALL</code><!-- /@w -->)</dt>
<dd><p>Full output (includes informational messages).
</p></dd>
</dl>
</dd>
<dt><code class="code">scale (default: 16)</code></dt>
<dd><p>Scaling option. The values can be combined with the bitwise OR operator and
may be the following:
</p>
<dl class="table">
<dt>1 (<code class="code">GLP_SF_GM</code><!-- /@w -->)</dt>
<dd><p>Geometric mean scaling.
</p>
</dd>
<dt>16 (<code class="code">GLP_SF_EQ</code><!-- /@w -->)</dt>
<dd><p>Equilibration scaling.
</p>
</dd>
<dt>32 (<code class="code">GLP_SF_2N</code><!-- /@w -->)</dt>
<dd><p>Round scale factors to power of two.
</p>
</dd>
<dt>64 (<code class="code">GLP_SF_SKIP</code><!-- /@w -->)</dt>
<dd><p>Skip if problem is well scaled.
</p></dd>
</dl>
<p>Alternatively, a value of 128 (<code class="env">GLP_SF_AUTO</code><!-- /@w -->) may be also
specified, in which case the routine chooses the scaling options
automatically.
</p>
</dd>
<dt><code class="code">dual (default: 1)</code></dt>
<dd><p>Simplex method option:
</p>
<dl class="table">
<dt>1 (<code class="code">GLP_PRIMAL</code><!-- /@w -->)</dt>
<dd><p>Use two-phase primal simplex.
</p>
</dd>
<dt>2 (<code class="code">GLP_DUALP</code><!-- /@w -->)</dt>
<dd><p>Use two-phase dual simplex, and if it fails, switch to the primal simplex.
</p>
</dd>
<dt>3 (<code class="code">GLP_DUAL</code><!-- /@w -->)</dt>
<dd><p>Use two-phase dual simplex.
</p></dd>
</dl>
</dd>
<dt><code class="code">price (default: 34)</code></dt>
<dd><p>Pricing option (for both primal and dual simplex):
</p>
<dl class="table">
<dt>17 (<code class="code">GLP_PT_STD</code><!-- /@w -->)</dt>
<dd><p>Textbook pricing.
</p>
</dd>
<dt>34 (<code class="code">GLP_PT_PSE</code><!-- /@w -->)</dt>
<dd><p>Steepest edge pricing.
</p></dd>
</dl>
</dd>
<dt><code class="code">itlim (default: intmax)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">outfrq (default: 200)</code></dt>
<dd><p>Output frequency, in iterations. This parameter specifies how frequently
the solver sends information about the solution to the standard output.
</p>
</dd>
<dt><code class="code">branch (default: 4)</code></dt>
<dd><p>Branching technique option (for MIP only):
</p>
<dl class="table">
<dt>1 (<code class="code">GLP_BR_FFV</code><!-- /@w -->)</dt>
<dd><p>First fractional variable.
</p>
</dd>
<dt>2 (<code class="code">GLP_BR_LFV</code><!-- /@w -->)</dt>
<dd><p>Last fractional variable.
</p>
</dd>
<dt>3 (<code class="code">GLP_BR_MFV</code><!-- /@w -->)</dt>
<dd><p>Most fractional variable.
</p>
</dd>
<dt>4 (<code class="code">GLP_BR_DTH</code><!-- /@w -->)</dt>
<dd><p>Heuristic by Driebeck and Tomlin.
</p>
</dd>
<dt>5 (<code class="code">GLP_BR_PCH</code><!-- /@w -->)</dt>
<dd><p>Hybrid pseudocost heuristic.
</p></dd>
</dl>
</dd>
<dt><code class="code">btrack (default: 4)</code></dt>
<dd><p>Backtracking technique option (for MIP only):
</p>
<dl class="table">
<dt>1 (<code class="code">GLP_BT_DFS</code><!-- /@w -->)</dt>
<dd><p>Depth first search.
</p>
</dd>
<dt>2 (<code class="code">GLP_BT_BFS</code><!-- /@w -->)</dt>
<dd><p>Breadth first search.
</p>
</dd>
<dt>3 (<code class="code">GLP_BT_BLB</code><!-- /@w -->)</dt>
<dd><p>Best local bound.
</p>
</dd>
<dt>4 (<code class="code">GLP_BT_BPH</code><!-- /@w -->)</dt>
<dd><p>Best projection heuristic.
</p></dd>
</dl>
</dd>
<dt><code class="code">presol (default: 1)</code></dt>
<dd><p>If this flag is set, the simplex solver uses the built-in LP presolver.
Otherwise the LP presolver is not used.
</p>
</dd>
<dt><code class="code">lpsolver (default: 1)</code></dt>
<dd><p>Select which solver to use. If the problem is a MIP problem this flag
will be ignored.
</p>
<dl class="table">
<dt>1</dt>
<dd><p>Revised simplex method.
</p>
</dd>
<dt>2</dt>
<dd><p>Interior point method.
</p></dd>
</dl>
</dd>
<dt><code class="code">rtest (default: 34)</code></dt>
<dd><p>Ratio test technique:
</p>
<dl class="table">
<dt>17 (<code class="code">GLP_RT_STD</code><!-- /@w -->)</dt>
<dd><p>Standard ("textbook").
</p>
</dd>
<dt>34 (<code class="code">GLP_RT_HAR</code><!-- /@w -->)</dt>
<dd><p>Harris’ two-pass ratio test.
</p></dd>
</dl>
</dd>
<dt><code class="code">tmlim (default: intmax)</code></dt>
<dd><p>Searching time limit, in milliseconds.
</p>
</dd>
<dt><code class="code">outdly (default: 0)</code></dt>
<dd><p>Output delay, in seconds. This parameter specifies how long the solver
should delay sending information about the solution to the standard output.
</p>
</dd>
<dt><code class="code">save (default: 0)</code></dt>
<dd><p>If this parameter is nonzero, save a copy of the problem in CPLEX
LP format to the file <samp class="file">"outpb.lp"</samp>. There is currently no way to
change the name of the output file.
</p></dd>
</dl>
<p>Real parameters:
</p>
<dl class="table">
<dt><code class="code">tolbnd (default: 1e-7)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">toldj (default: 1e-7)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">tolpiv (default: 1e-9)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">objll (default: -DBL_MAX)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">objul (default: +DBL_MAX)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">tolint (default: 1e-5)</code></dt>
<dd><p>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.
</p>
</dd>
<dt><code class="code">tolobj (default: 1e-7)</code></dt>
<dd><p>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.
</p></dd>
</dl>
</dd>
</dl>
<p>Output values:
</p>
<dl class="table">
<dt><var class="var">xopt</var></dt>
<dd><p>The optimizer (the value of the decision variables at the optimum).
</p>
</dd>
<dt><var class="var">fopt</var></dt>
<dd><p>The optimum value of the objective function.
</p>
</dd>
<dt><var class="var">errnum</var></dt>
<dd><p>Error code.
</p>
<dl class="table">
<dt>0</dt>
<dd><p>No error.
</p>
</dd>
<dt>1 (<code class="code">GLP_EBADB</code><!-- /@w -->)</dt>
<dd><p>Invalid basis.
</p>
</dd>
<dt>2 (<code class="code">GLP_ESING</code><!-- /@w -->)</dt>
<dd><p>Singular matrix.
</p>
</dd>
<dt>3 (<code class="code">GLP_ECOND</code><!-- /@w -->)</dt>
<dd><p>Ill-conditioned matrix.
</p>
</dd>
<dt>4 (<code class="code">GLP_EBOUND</code><!-- /@w -->)</dt>
<dd><p>Invalid bounds.
</p>
</dd>
<dt>5 (<code class="code">GLP_EFAIL</code><!-- /@w -->)</dt>
<dd><p>Solver failed.
</p>
</dd>
<dt>6 (<code class="code">GLP_EOBJLL</code><!-- /@w -->)</dt>
<dd><p>Objective function lower limit reached.
</p>
</dd>
<dt>7 (<code class="code">GLP_EOBJUL</code><!-- /@w -->)</dt>
<dd><p>Objective function upper limit reached.
</p>
</dd>
<dt>8 (<code class="code">GLP_EITLIM</code><!-- /@w -->)</dt>
<dd><p>Iterations limit exhausted.
</p>
</dd>
<dt>9 (<code class="code">GLP_ETMLIM</code><!-- /@w -->)</dt>
<dd><p>Time limit exhausted.
</p>
</dd>
<dt>10 (<code class="code">GLP_ENOPFS</code><!-- /@w -->)</dt>
<dd><p>No primal feasible solution.
</p>
</dd>
<dt>11 (<code class="code">GLP_ENODFS</code><!-- /@w -->)</dt>
<dd><p>No dual feasible solution.
</p>
</dd>
<dt>12 (<code class="code">GLP_EROOT</code><!-- /@w -->)</dt>
<dd><p>Root LP optimum not provided.
</p>
</dd>
<dt>13 (<code class="code">GLP_ESTOP</code><!-- /@w -->)</dt>
<dd><p>Search terminated by application.
</p>
</dd>
<dt>14 (<code class="code">GLP_EMIPGAP</code><!-- /@w -->)</dt>
<dd><p>Relative MIP gap tolerance reached.
</p>
</dd>
<dt>15 (<code class="code">GLP_ENOFEAS</code><!-- /@w -->)</dt>
<dd><p>No primal/dual feasible solution.
</p>
</dd>
<dt>16 (<code class="code">GLP_ENOCVG</code><!-- /@w -->)</dt>
<dd><p>No convergence.
</p>
</dd>
<dt>17 (<code class="code">GLP_EINSTAB</code><!-- /@w -->)</dt>
<dd><p>Numerical instability.
</p>
</dd>
<dt>18 (<code class="code">GLP_EDATA</code><!-- /@w -->)</dt>
<dd><p>Invalid data.
</p>
</dd>
<dt>19 (<code class="code">GLP_ERANGE</code><!-- /@w -->)</dt>
<dd><p>Result out of range.
</p></dd>
</dl>
</dd>
<dt><var class="var">extra</var></dt>
<dd><p>A data structure containing the following fields:
</p>
<dl class="table">
<dt><code class="code">lambda</code></dt>
<dd><p>Dual variables.
</p>
</dd>
<dt><code class="code">redcosts</code></dt>
<dd><p>Reduced Costs.
</p>
</dd>
<dt><code class="code">time</code></dt>
<dd><p>Time (in seconds) used for solving LP/MIP problem.
</p>
</dd>
<dt><code class="code">status</code></dt>
<dd><p>Status of the optimization.
</p>
<dl class="table">
<dt>1 (<code class="code">GLP_UNDEF</code><!-- /@w -->)</dt>
<dd><p>Solution status is undefined.
</p>
</dd>
<dt>2 (<code class="code">GLP_FEAS</code><!-- /@w -->)</dt>
<dd><p>Solution is feasible.
</p>
</dd>
<dt>3 (<code class="code">GLP_INFEAS</code><!-- /@w -->)</dt>
<dd><p>Solution is infeasible.
</p>
</dd>
<dt>4 (<code class="code">GLP_NOFEAS</code><!-- /@w -->)</dt>
<dd><p>Problem has no feasible solution.
</p>
</dd>
<dt>5 (<code class="code">GLP_OPT</code><!-- /@w -->)</dt>
<dd><p>Solution is optimal.
</p>
</dd>
<dt>6 (<code class="code">GLP_UNBND</code><!-- /@w -->)</dt>
<dd><p>Problem has no unbounded solution.
</p></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
<p>Example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">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);
</pre></div></div>
</dd></dl>
</div>
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