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diff --git a/help/en_US/intqpipopt.xml b/help/en_US/intqpipopt.xml new file mode 100644 index 0000000..2859568 --- /dev/null +++ b/help/en_US/intqpipopt.xml @@ -0,0 +1,403 @@ +<?xml version="1.0" encoding="UTF-8"?> + +<!-- + * + * This help file was generated from intqpipopt.sci using help_from_sci(). + * + --> + +<refentry version="5.0-subset Scilab" xml:id="intqpipopt" 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>intqpipopt</refname> + <refpurpose>Solves a linear quadratic problem.</refpurpose> + </refnamediv> + + +<refsynopsisdiv> + <title>Calling Sequence</title> + <synopsis> + xopt = intqpipopt(H,f) + xopt = intqpipopt(H,f,intcon) + xopt = intqpipopt(H,f,intcon,A,b) + xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq) + xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub) + xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0) + xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options) + xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options,"file_path") + [xopt,fopt,exitflag,output] = intqpipopt( ... ) + + </synopsis> +</refsynopsisdiv> + +<refsection> + <title>Input Parameters</title> + <variablelist> + <varlistentry><term>H :</term> + <listitem><para> A symmetric matrix of doubles, representing the Hessian of the quadratic problem.</para></listitem></varlistentry> + <varlistentry><term>f :</term> + <listitem><para> A vector of doubles, representing coefficients of the linear terms in the quadratic problem.</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>A :</term> + <listitem><para> A matrix of doubles, containing the coefficients of linear inequality constraints of size (m X n) where 'm' is the number of linear inequality constraints.</para></listitem></varlistentry> + <varlistentry><term>b :</term> + <listitem><para> A vector of doubles, related to 'A' and represents the linear coefficients in the linear inequality constraints of size (m X 1).</para></listitem></varlistentry> + <varlistentry><term>Aeq :</term> + <listitem><para> A matrix of doubles, containing the coefficients of linear equality constraints of size (m1 X n) where 'm1' is the number of linear equality constraints.</para></listitem></varlistentry> + <varlistentry><term>beq :</term> + <listitem><para> A vector of double, vector of doubles, related to 'Aeq' and represents the linear coefficients in the equality constraints of size (m1 X 1).</para></listitem></varlistentry> + <varlistentry><term>lb :</term> + <listitem><para> A vector of doubles, containing the lower bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.</para></listitem></varlistentry> + <varlistentry><term>ub :</term> + <listitem><para> A vector of doubles, containing the upper bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.</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>options :</term> + <listitem><para> A list, containing the option for user to specify. See below for details.</para></listitem></varlistentry> + <varlistentry><term>file_path :</term> + <listitem><para> path to bonmin opt file if used.</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 value of the function at xopt.</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>output :</term> + <listitem><para> A structure, containing the information about the optimization. See below for details.</para></listitem></varlistentry> +</variablelist> +</refsection> + +<refsection> + <title>Description</title> + <para> +Search the minimum of a constrained linear quadratic optimization problem specified by : + </para> + <para> +<latex> +\begin{eqnarray} +&\mbox{min}_{x} +& 1/2⋅x^T⋅H⋅x + f^T⋅x \\ +& \text{subject to} & A⋅x \leq b \\ +& & Aeq⋅x = beq \\ +& & lb \leq x \leq ub \\ +& & x_i \in \!\, \mathbb{Z}, i \in \!\, intcon\\ +\end{eqnarray} +</latex> + </para> + <para> +intqpipopt calls Bonmin, a library written in C++ to solve the quadratic problem. + </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> + <para> +The output data structure contains detailed information about the optimization process. +It is of type "struct" and contains the following fields. +<itemizedlist> +<listitem>output.constrviolation: The max-norm of the constraint violation.</listitem> +</itemizedlist> + </para> + <para> +</para> +</refsection> +<para> +A few examples displaying the various functionalities of intqpipopt 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> + Here we solve a simple objective function. + </para> + <para> +Find x in R^6 such that it minimizes: + </para> + <para> +<latex> +\begin{eqnarray} +\mbox{min}_{x}\ f(x) &= \dfrac{1}{2}x'\boldsymbol{\cdot} H\boldsymbol{\cdot}x + f' \boldsymbol{\cdot} x\\ +\text{Where: } H &= I_{6}\\ +F &= +\begin{array}{cccccc} +[1 & 2 & 3 & 4 & 5 & 6] +\end{array} +\end{eqnarray}\\ +\text{With integer constraints as: } \\ +\begin{eqnarray} +\begin{array}{c} +[2 & 4] \\ +\end{array} +\end{eqnarray} +</latex> + </para> + <para> + + </para> + <programlisting role="example"><![CDATA[ +//Example 1: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=qpipoptmat(H,f) + ]]></programlisting> +</refsection> + +<refsection> + <title>Example</title> + We proceed to add simple linear inequality constraints. + +<para> +<latex> +\begin{eqnarray} +\&x_{2} + x_{4}+ 2x_{5} - x_{6}&\leq -1\\ +\&-x_{1} + 2x_{3} + x_{4} + x_{5}&\leq 2.5\\ +\end{eqnarray} +</latex> + </para> + <para> + + </para> + <programlisting role="example"><![CDATA[ +//Example 2: +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b) + ]]></programlisting> +</refsection> + +<refsection> + <title>Example</title> + Here we build up on the previous example by adding linear equality constraints. +We add the following constraints to the problem specified above: +<para> +<latex> +\begin{eqnarray} +&x_{1} - x_{2} + x_{3} + 3x_{5} + x_{6}&= 1 \\ +&-x_{1} + 2x_{3}+ x_{4} + x_{5}&= 2\\ +&2x_{1} + 5x_{2}+ 3x_{3} + x_{5}&= 3 +\end{eqnarray} +</latex> + </para> + <para> + + </para> + <programlisting role="example"><![CDATA[ +//Example 3: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Equality constraints +Aeq= [1,-1,1,0,3,1; +-1,0,-3,-4,5,6; +2,5,3,0,1,0]; +beq=[1; 2; 3]; +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq) + ]]></programlisting> +</refsection> + +<refsection> +<title>Example</title> +<para> + In this example, we proceed to add the upper and lower bounds to the objective function. +</para> +<para> +<latex> +\begin{eqnarray} +-1000 &\leq x_{1} &\leq 10000\\ +-10000 &\leq x_{2} &\leq 100\\ +0 &\leq x_{3} &\leq 1.5\\ +-1000 &\leq x_{4} &\leq 100\\ +-1000 &\leq x_{5} &\leq 100\\ +-1000 &\leq x_{6} &\leq 1000 +\end{eqnarray} +</latex> + </para> + <para> + + </para> + <programlisting role="example"><![CDATA[ +//Example 4: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Equality constraints +Aeq= [1,-1,1,0,3,1; +-1,0,-3,-4,5,6; +2,5,3,0,1,0]; +beq=[1; 2; 3]; +//Variable bounds +lb=[-1000; -10000; 0; -1000; -1000; -1000]; +ub=[10000; 100; 1.5; 100; 100; 1000]; +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub) + ]]></programlisting> +</refsection> + +<refsection> + <title>Example</title> +<para> +In this example, we initialize the values of x to speed up the computation. We further enhance the functionality of qpipoptmat by setting input options. +</para> + <programlisting role="example"><![CDATA[ +//Example 5: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Equality constraints +Aeq= [1,-1,1,0,3,1; +-1,0,-3,-4,5,6; +2,5,3,0,1,0]; +beq=[1; 2; 3]; +//Variable bounds +lb=[-1000; -10000; 0; -1000; -1000; -1000]; +ub=[10000; 100; 1.5; 100; 100; 1000]; +//Initial guess and options +x0 = repmat(0,6,1); +options = list("MaxIter", 300, "CpuTime", 100); +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options) + ]]></programlisting> +</refsection> + +<refsection> +<title>Example</title> +Infeasible Problems: Find x in R^6 such that it minimizes the objective function used above under the following constraints: +<para> +<latex> +\begin{eqnarray} +begin{eqnarray} +\hspace{70pt} &x_{2} + x_{4}+ 2x_{5} - x_{6}&\leq -1\\ +\hspace{70pt} &-x_{1} + 2x_{3} + x_{4} + x_{5}&\leq 2.5\\ +\hspace{70pt} &x_{2} + x_{4}+ 2x_{5} - x_{6}&= 4 \\ +\hspace{70pt} &-x_{1} + 2x_{3}+ x_{4} + x_{5}&= 2\\ +\\ \end{eqnarray}\\ +\text{With integer constraints as: } \\ +\begin{eqnarray} +\begin{array}{c} +[2 & 4] \\ +\end{array} +\end{eqnarray} +</latex> + </para> +<para> +</para> +<programlisting role="example"><![CDATA[ +//Example 6: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Equality constraints +Aeq= [0,1,0,1,2,-1; +-1,0,-3,-4,5,6]; +beq=[4; 2]; +//Integer Constraints +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq) +]]></programlisting> +</refsection> + +<refsection> + <title>Example</title> +<para> +Unbounded Problems: Find x in R^6 such that it minimizes the objective function used above under the following constraints: +</para> +<para> +<latex> +\begin{eqnarray} +\mbox{min}_{x}\ f(x) &= \dfrac{1}{2}x'\boldsymbol{\cdot} H\boldsymbol{\cdot}x + f' \boldsymbol{\cdot} x\\ +\text{Where H is specified below and} \\ +F &= +\begin{array}{cccccc} +[1 & 2 & 3 & 4 & 5 & 6] +\end{array} +\end{eqnarray}\\ +\text{Subjected to: }\\ +\begin{eqnarray} +\hspace{70pt} &x_{2} + x_{4}+ 2x_{5} - x_{6}&\leq -1\\ +\hspace{70pt} &-x_{1} + 2x_{3} + x_{4} + x_{5}&\leq 2.5\\ +\hspace{70pt} &x_{1} - x_{2} + x_{3} + 3x_{5} + x_{6}&= 1 \\ +\hspace{70pt} &-x_{1} + 2x_{3}+ x_{4} + x_{5}&= 2\\ +\\ \end{eqnarray} +</latex> + </para> +<para> +</para> + <programlisting role="example"><![CDATA[ +//Example 7: +//Minimize 0.5*x'*H*x + f'*x with +f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); H(1,1) = -1; +//Inequality constraints +A= [0,1,0,1,2,-1; +-1,0,2,1,1,0]; +b = [-1; 2.5]; +//Equality constraints +Aeq= [1,-1,1,0,3,1; +-1,0,-3,-4,5,6]; +beq=[1; 2]; +intcon = [2 ,4]; +//Running intqpipopt +[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq) +]]></programlisting> +</refsection> + +<refsection> + <title>Authors</title> + <simplelist type="vert"> + <member>Akshay Miterani and Pranav Deshpande</member> + </simplelist> +</refsection> +</refentry> |