Quickguide MIQQ Problem

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

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

${\begin{array}{ccccccl}\min \limits _{x}&{5}{l}{f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x}\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&&&x^{T}Q^{(i)}x+a^{(i)T}x&\leq &r_{U}^{(i)},&i=1,\ldots ,n_{qc}\\&&&x_{i}\mathrm {\ \ integer} &&i\in I\\\end{array}}$

where $c,x,x_{L},x_{U},a^{(i)}\in \mathbb {R} ^{n}$, $F,Q^{(i)}\in \mathbb {R} ^{n\times n}$, $A\in \mathbb {R} ^{m\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m}$. $r_{U}^{(i)}$ is a scalar. The variables $x\in I$, the index subset of $1,...,n$, are restricted to be integers.

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

File: tomlab/quickguide/miqqQG.m

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

 % miqqQG is a small example problem for defining and solving
 % mixed-integer quadratic programming problems with quadratic constraints 
 % using the TOMLAB format.
 
 Name = 'MIQQ Test Problem 1';
 f_Low = -1E5;
 x_opt = [];
 f_opt = [];
 IntVars = logical([0 0 1]); % 3rd variable is integer valued
 
 F   = [2 0 0;0 2 0;0 0 2];
 A   = [1 2 -1;1 -1 1];
 b_L = [4 -2]';
 b_U = b_L;
 c   = zeros(3,1);
 
 x_0 = [0 0 0]';
 x_L = [-10 -10 -10]';
 x_U = [10 10 10]';
 x_min = [0 0 -1]';
 x_max = [2 2 1]';
 
 % Adding quadratic constraints
 clear qc
 qc(1).Q = speye(3,3);
 qc(1).a = zeros(3,1);
 qc(1).r_U = 3;
 
 qc(2).Q = speye(3,3);
 qc(2).a = zeros(3,1);
 qc(2).r_U = 5;
 
 Prob = miqqAssign(F, c, A, b_L, b_U, x_L, x_U, x_0, qc,...
                   IntVars, [], [], [],...
                   Name, [], [],...
                   x_min, x_max, f_opt, x_opt);
 
 Result = tomRun('cplex', Prob, 1);
 % Result = tomRun('minlpBB', Prob, 1);