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<?xml version="1.0" encoding="UTF-8"?>
<!--
*
* This help file was generated from intfminunc.sci using help_from_sci().
*
-->
<refentry version="5.0-subset Scilab" xml:id="intfminunc" xml:lang="en"
xmlns="http://docbook.org/ns/docbook"
xmlns:xlink="http://www.w3.org/1999/xlink"
xmlns:svg="http://www.w3.org/2000/svg"
xmlns:ns3="http://www.w3.org/1999/xhtml"
xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:scilab="http://www.scilab.org"
xmlns:db="http://docbook.org/ns/docbook">
<refnamediv>
<refname>intfminunc</refname>
<refpurpose>Solves an unconstrainted multi-variable mixed integer non linear programming optimization problem</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>
xopt = intfminunc(f,x0)
xopt = intfminunc(f,x0,intcon)
xopt = intfminunc(f,x0,intcon,options)
[xopt,fopt] = intfminunc(.....)
[xopt,fopt,exitflag]= intfminunc(.....)
[xopt,fopt,exitflag,gradient,hessian]= intfminunc(.....)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Input Parameters</title>
<variablelist>
<varlistentry><term>f :</term>
<listitem><para> A function, representing the objective function of the problem.</para></listitem></varlistentry>
<varlistentry><term>x0 :</term>
<listitem><para> A vector of doubles, containing the starting values of variables of size (1 X n) or (n X 1) where 'n' is the number of Variables.</para></listitem></varlistentry>
<varlistentry><term>intcon :</term>
<listitem><para> A vector of integers, representing the variables that are constrained to be integers.</para></listitem></varlistentry>
<varlistentry><term>options :</term>
<listitem><para> A list, containing the option for user to specify. See below for details.</para></listitem></varlistentry>
</variablelist>
</refsection>
<refsection>
<title> Outputs</title>
<variablelist>
<varlistentry><term>xopt :</term>
<listitem><para> A vector of doubles, containing the computed solution of the optimization problem.</para></listitem></varlistentry>
<varlistentry><term>fopt :</term>
<listitem><para> A double, containing the the function value at x.</para></listitem></varlistentry>
<varlistentry><term>exitflag :</term>
<listitem><para> An integer, containing the flag which denotes the reason for termination of algorithm. See below for details.</para></listitem></varlistentry>
<varlistentry><term>gradient :</term>
<listitem><para> A vector of doubles, containing the objective's gradient of the solution.</para></listitem></varlistentry>
<varlistentry><term>hessian :</term>
<listitem><para> A matrix of doubles, containing the Lagrangian's hessian of the solution.</para></listitem></varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Search the minimum of a multi-variable mixed integer non linear programming unconstrained optimization problem specified by :
Find the minimum of f(x) such that
</para>
<para>
<latex>
\begin{eqnarray}
&\mbox{min}_{x}
& f(x)
& x_{i} \in \!\, \mathbb{Z}, i \in \!\, I
\end{eqnarray}
</latex>
</para>
<para>
intfminunc calls Bonmin, which is an optimization library written in C++, to solve the bound optimization problem.
</para>
<para>
<title>Options</title>
The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:
</para>
<para>
options= list("IntegerTolerance", [---], "MaxNodes",[---], "MaxIter", [---], "AllowableGap",[---] "CpuTime", [---],"gradobj", "off", "hessian", "off" );
<itemizedlist>
<listitem>IntegerTolerance : A Scalar, a number with that value of an integer is considered integer.</listitem>
<listitem>MaxNodes : A Scalar, containing the maximum number of nodes that the solver should search.</listitem>
<listitem>CpuTime : A scalar, specifying the maximum amount of CPU Time in seconds that the solver should take.</listitem>
<listitem>AllowableGap : A scalar, that specifies the gap between the computed solution and the the objective value of the best known solution stop, at which the tree search can be stopped.</listitem>
<listitem>MaxIter : A scalar, specifying the maximum number of iterations that the solver should take.</listitem>
<listitem>gradobj : A string, to turn on or off the user supplied objective gradient.</listitem>
<listitem>hessian : A scalar, to turn on or off the user supplied objective hessian.</listitem>
</itemizedlist>
The default values for the various items are given as:
</para>
<para>
options = list('integertolerance',1d-06,'maxnodes',2147483647,'cputime',1d10,'allowablegap',0,'maxiter',2147483647,'gradobj',"off",'hessian',"off")
</para>
<para>
</para>
<para>
The exitflag allows to know the status of the optimization which is given back by Ipopt.
<itemizedlist>
<listitem>0 : Optimal Solution Found </listitem>
<listitem>1 : InFeasible Solution.</listitem>
<listitem>2 : Objective Function is Continuous Unbounded.</listitem>
<listitem>3 : Limit Exceeded.</listitem>
<listitem>4 : User Interrupt.</listitem>
<listitem>5 : MINLP Error.</listitem>
</itemizedlist>
</para>
<para>
For more details on exitflag, see the Bonmin documentation which can be found on http://www.coin-or.org/Bonmin
</para>
<para>
</para>
</refsection>
<para>
A few examples displaying the various functionalities of intfminunc have been provided below. You will find a series of problems and the appropriate code snippets to solve them.
</para>
<refsection>
<title>Example</title>
<para>
We begin with the minimization of a simple non-linear function.
</para>
<para>
Find x in R^2 such that it minimizes:
</para>
<para>
<latex>
\begin{eqnarray}
\mbox{min}_{x}\ f(x) = x_{1}^{2} + x_{2}^{2}
\end{eqnarray}\\
\text{With integer constraints as: } \\
\begin{eqnarray}
\begin{array}{c}
[1] \\
\end{array}
\end{eqnarray}
</latex>
</para>
<para>
</para>
<programlisting role="example"><![CDATA[
//Example 1:
//Objective function to be minimised
function y= f(x)
y= x(1)^2 + x(2)^2;
endfunction
//Starting point
x0=[2,1];
intcon = [1];
[xopt,fopt]=intfminunc(f,x0,intcon)
// Press ENTER to continue
]]></programlisting>
</refsection>
<refsection>
<title>Example</title>
<para>
We now look at the Rosenbrock function, a non-convex performance test problem for optimization routines. We use this example to illustrate how we can enhance the functionality of intfminunc by setting input options. We can pre-define the gradient of the objective function and/or the hessian of the lagrange function and thereby improve the speed of computation. This is elaborated on in example 2. We also set solver parameters using the options.
</para>
<para>
<latex>
\begin{eqnarray}
\mbox{min}_{x}\ f(x) = 100\boldsymbol{\cdot} (x_{2} - x_{1}^{2})^{2} + (1-x_{1})^{2}
\end{eqnarray}\\
\text{With integer constraints as: } \\
\begin{eqnarray}
\begin{array}{c}
[2] \\
\end{array}
\end{eqnarray}
</latex>
</para>
<para>
</para>
<programlisting role="example"><![CDATA[
///Example 2:
//Objective function to be minimised
function y= f(x)
y= 100*(x(2) - x(1)^2)^2 + (1-x(1))^2;
endfunction
//Starting point
x0=[-1,2];
intcon = [2]
//Options
options=list("MaxIter", [1500], "CpuTime", [500]);
//Calling
[xopt,fopt,exitflag,gradient,hessian]=intfminunc(f,x0,intcon,options)
// Press ENTER to continue
]]></programlisting>
</refsection>
<refsection>
<title>Example</title>
<para>
Unbounded Problems: Find x in R^2 such that it minimizes:
</para>
<para>
<latex>
\begin{eqnarray}
f(x) = -x_{1}^{2} - x_{2}^{2}
\end{eqnarray}\\
\text{With integer constraints as: } \\
\begin{eqnarray}
\begin{array}{c}
[1] \\
\end{array}
\end{eqnarray}
</latex>
</para>
<para>
</para>
<programlisting role="example"><![CDATA[
//The below problem is an unbounded problem:
//Find x in R^2 such that the below function is minimum
//f = - x1^2 - x2^2
//Objective function to be minimised
function [y,g,h] = f(x)
y = -x(1)^2 - x(2)^2;
g = [-2*x(1),-2*x(2)];
h = [-2,0;0,-2];
endfunction
//Starting point
x0=[2,1];
intcon = [1]
options = list("gradobj","ON","hessian","on");
[xopt,fopt,exitflag,gradient,hessian]=intfminunc(f,x0,intcon,options)
]]></programlisting>
</refsection>
</refentry>
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