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<a name="Functions-and-Variables-for-simplex"></a>
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<p>
Previous: <a href="maxima_354.html#Introduction-to-simplex" accesskey="p" rel="previous">Introduction to simplex</a>, Up: <a href="maxima_353.html#simplex_002dpkg" accesskey="u" rel="up">simplex-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-simplex-1"></a>
<h3 class="section">86.2 Functions and Variables for simplex</h3>
<a name="epsilon_005flp"></a><a name="Item_003a-simplex_002fdefvr_002fepsilon_005flp"></a><dl>
<dt><a name="index-epsilon_005flp"></a>Option variable: <strong>epsilon_lp</strong></dt>
<dd><p>Default value: <code>10^-8</code>
</p>
<p>Epsilon used for numerical computations in <code>linear_program</code>; it is
set to 0 in <code>linear_program</code> when all inputs are rational.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) load("simplex")$
(%i2) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]);
Warning: linear_program(A,b,c): non-rat inputs found, epsilon_lp= 1.0e-8
Warning: Solution may be incorrect.
(%o2) Problem not bounded!
(%i3) minimize_lp(-x, [10^-9*x + y <= 1], [x,y]);
(%o3) [- 1000000000, [y = 0, x = 1000000000]]
(%i4) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]), epsilon_lp=0;
(%o4) [- 9.999999999999999e+8, [y = 0, x = 9.999999999999999e+8]]
</pre></div>
<p>See also: <code><a href="#linear_005fprogram">linear_program</a></code>, <code><a href="maxima_13.html#ratnump">ratnump</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·</div>
</dd></dl>
<a name="linear_005fprogram"></a><a name="Item_003a-simplex_002fdeffn_002flinear_005fprogram"></a><dl>
<dt><a name="index-linear_005fprogram"></a>Function: <strong>linear_program</strong> <em>(<var>A</var>, <var>b</var>, <var>c</var>)</em></dt>
<dd>
<p><code>linear_program</code> is an implementation of the simplex algorithm.
<code>linear_program(A, b, c)</code> computes a vector <var>x</var> for which
<code>c.x</code> is minimum possible among vectors for which <code>A.x = b</code>
and <code>x >= 0</code>. Argument <var>A</var> is a matrix and arguments <var>b</var>
and <var>c</var> are lists.
</p>
<p><code>linear_program</code> returns a list which contains the minimizing
vector <var>x</var> and the minimum value <code>c.x</code>. If the problem is not
bounded, it returns "Problem not bounded!" and if the problem is not
feasible, it returns "Problem not feasible!".
</p>
<p>To use this function first load the <code>simplex</code> package with
<code>load("simplex");</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
(%i3) b: [1,1,6]$
(%i4) c: [1,-2,0,0]$
(%i5) linear_program(A, b, c);
13 19 3
(%o5) [[--, 4, --, 0], - -]
2 2 2
</pre></div>
<p>See also: <code><a href="#minimize_005flp">minimize_lp</a></code>, <code><a href="#scale_005flp">scale_lp</a></code>, and <code><a href="#epsilon_005flp">epsilon_lp</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·<a href="maxima_424.html#Category_003a-Numerical-methods">Numerical methods</a>
·</div>
</dd></dl>
<a name="maximize_005flp"></a><a name="Item_003a-simplex_002fdeffn_002fmaximize_005flp"></a><dl>
<dt><a name="index-maximize_005flp"></a>Function: <strong>maximize_lp</strong> <em>(<var>obj</var>, <var>cond</var>, [<var>pos</var>])</em></dt>
<dd>
<p>Maximizes linear objective function <var>obj</var> subject to some linear
constraints <var>cond</var>. See <code><a href="#minimize_005flp">minimize_lp</a></code> for detailed
description of arguments and return value.
</p>
<p>See also: <code><a href="#minimize_005flp">minimize_lp</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·<a href="maxima_424.html#Category_003a-Numerical-methods">Numerical methods</a>
·</div>
</dd></dl>
<a name="minimize_005flp"></a><a name="Item_003a-simplex_002fdeffn_002fminimize_005flp"></a><dl>
<dt><a name="index-minimize_005flp"></a>Function: <strong>minimize_lp</strong> <em>(<var>obj</var>, <var>cond</var>, [<var>pos</var>])</em></dt>
<dd>
<p>Minimizes a linear objective function <var>obj</var> subject to some linear
constraints <var>cond</var>. <var>cond</var> a list of linear equations or
inequalities. In strict inequalities <code>></code> is replaced by <code>>=</code>
and <code><</code> by <code><=</code>. The optional argument <var>pos</var> is a list
of decision variables which are assumed to be positive.
</p>
<p>If the minimum exists, <code>minimize_lp</code> returns a list which
contains the minimum value of the objective function and a list of
decision variable values for which the minimum is attained. If the
problem is not bounded, <code>minimize_lp</code> returns "Problem not
bounded!" and if the problem is not feasible, it returns "Problem not
feasible!".
</p>
<p>The decision variables are not assumed to be non-negative by default. If
all decision variables are non-negative, set <code>nonnegative_lp</code> to
<code>true</code> or include <code>all</code> in the optional argument <var>pos</var>. If
only some of decision variables are positive, list them in the optional
argument <var>pos</var> (note that this is more efficient than adding
constraints).
</p>
<p><code>minimize_lp</code> uses the simplex algorithm which is implemented in
maxima <code>linear_program</code> function.
</p>
<p>To use this function first load the <code>simplex</code> package with
<code>load("simplex");</code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
4 6 2
(%o1) [-, [y = -, x = - -]]
5 5 5
(%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonnegative_lp=true;
(%o2) [1, [y = 1, x = 0]]
(%i3) minimize_lp(x+y, [3*x+y>0, x+2*y>2], all);
(%o3) [1, [y = 1, x = 0]]
(%i4) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonnegative_lp=true;
(%o4) Problem not feasible!
(%i5) minimize_lp(x+y, [3*x+y>0]);
(%o5) Problem not bounded!
</pre></div>
<p>There is also a limited ability to solve linear programs with symbolic
constants.
</p>
<div class="example">
<pre class="example">(%i1) declare(c,constant)$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
Is (c-1)*c positive, negative or zero?
p;
Is c*(2*c-1) positive, negative or zero?
p;
Is c positive or negative?
p;
Is c-1 positive, negative or zero?
p;
Is 2*c-1 positive, negative or zero?
p;
Is 3*c-4 positive, negative or zero?
p;
1 1
(%o2) [4, [x = -----, y = 3 - ---------]]
1 1
1 - - (1 - -) c
c c
</pre></div>
<div class="example">
<pre class="example">(%i1) (assume(c>4/3), declare(c,constant))$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
1 1
(%o2) [4, [x = -----, y = 3 - ---------]]
1 1
1 - - (1 - -) c
c c
</pre></div>
<p>See also: <code><a href="#maximize_005flp">maximize_lp</a></code>, <code><a href="#nonnegative_005flp">nonnegative_lp</a></code>, <code><a href="#epsilon_005flp">epsilon_lp</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·<a href="maxima_424.html#Category_003a-Numerical-methods">Numerical methods</a>
·</div>
</dd></dl>
<a name="nonnegative_005flp"></a><a name="Item_003a-simplex_002fdefvr_002fnonnegative_005flp"></a><dl>
<dt><a name="index-nonnegative_005flp"></a>Option variable: <strong>nonnegative_lp</strong></dt>
<dd><a name="Item_003a-simplex_002fdefvr_002fnonegative_005flp"></a></dd><dt><a name="index-nonegative_005flp"></a>Option variable: <strong>nonegative_lp</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>If <code>nonnegative_lp</code> is true all decision variables to
<code>minimize_lp</code> and <code>maximize_lp</code> are assumed to be non-negative.
<code>nonegative_lp</code> is a deprecated alias.
</p>
<p>See also: <code><a href="#minimize_005flp">minimize_lp</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·</div>
</dd></dl>
<a name="scale_005flp"></a><a name="Item_003a-simplex_002fdefvr_002fscale_005flp"></a><dl>
<dt><a name="index-scale_005flp"></a>Option variable: <strong>scale_lp</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>scale_lp</code> is <code>true</code>,
<code>linear_program</code> scales its input so that the maximum absolute value in each row or column is 1.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·</div>
</dd></dl>
<a name="pivot_005fcount_005fsx"></a><a name="Item_003a-simplex_002fdefvr_002fpivot_005fcount_005fsx"></a><dl>
<dt><a name="index-pivot_005fcount_005fsx"></a>Variable: <strong>pivot_count_sx</strong></dt>
<dd>
<p>After <code>linear_program</code> returns,
<code>pivot_count_sx</code> is the number of pivots in last computation.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·</div>
</dd></dl>
<a name="pivot_005fmax_005fsx"></a><a name="Item_003a-simplex_002fdefvr_002fpivot_005fmax_005fsx"></a><dl>
<dt><a name="index-pivot_005fmax_005fsx"></a>Variable: <strong>pivot_max_sx</strong></dt>
<dd>
<p><code>pivot_max_sx</code> is the maximum number of pivots allowed by <code>linear_program</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-simplex">Package simplex</a>
·</div>
</dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_354.html#Introduction-to-simplex" accesskey="p" rel="previous">Introduction to simplex</a>, Up: <a href="maxima_353.html#simplex_002dpkg" accesskey="u" rel="up">simplex-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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