Quickguide MIQP Problem

This page is part of the Quickguide Manual. See Quickguide.

The general formulation in TOMLAB for a mixed-integer quadratic programming problem is:

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x\\&\\s/t&{\begin{array}{lcccl}x_{L}&\leq &x&\leq &x_{U},\\b_{L}&\leq &Ax&\leq &b_{U},~x_{j}\in \mathbb {N} \ ~~\forall j\in \$I\$\\\end{array}}\end{array}}$

where $c,x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $A\in \mathbb {R} ^{m_{1}\times n}$, and $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$. The variables $x\in I$, the index subset of $1,...,n$ are restricted to be integers. Equality constraints are defined by setting the lower bound equal to the upper bound, i.e. for constraint $i$: $b_{L}(i)=b_{U}(i)$.

The following file illustrates how to solve a MIQP problem in TOMLAB.

File: tomlab/quickguide/miqpQG.m

Open the file for viewing, and execute miqpQG in Matlab.

 % miqpQG is a small example problem for defining and solving
 % mixed-integer quadratic programming problems using the TOMLAB format.
 
 c    = [-6 0]';
 Name = 'XP Ref Manual MIQP';
 F    = [4 -2;-2 4];
 A    = [1 1];
 b_L  = -Inf;
 b_U  = 1.9;
 x_L  = [0 0]';
 x_U  = [Inf Inf]';
 
 % Defining first variable as an integer
 IntVars   = 1;
 
 % Assign routine for defining a MIQP problem.
 Prob = miqpAssign(F, c, A, b_L, b_U, x_L, x_U, [], ...
            IntVars, [], [], [], Name, [], []);
 
 % Calling driver routine tomRun to run the solver.
 % The 1 sets the print level after optimization.
 
 Result = tomRun('cplex', Prob, 1);
 %Result = tomRun('oqnlp', Prob, 1);
 %Result = tomRun('miqpBB', Prob, 1);
 %Result = tomRun('xpress-mp', Prob, 1);
 %Result = tomRun('minlpBB', Prob, 1);