Models Mixed-Integer Quadratic Programming Problems with Quadratic Constraints: miqq prob

This page is part of TOMLAB Models. See TOMLAB Models.

In miqq_prob there are 14 mixed-integer quadratic programming test problems with quadratic constraints with sizes to 10 variables and 8 constraints. In order to define the problem n and solve it execute the following in Matlab:

Prob	= probInit('miqq_prob',n); 
Result  = tomRun('',Prob);

The basic structure of a general mixed-integer quadratic programming problem with quadratic constraints is:

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

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.

An example of a problem of this class, (that is also found in the TOMLAB quickguide) is miqqQG:

File: tomlab/quickguide/miqqQG.m

% 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 = [0 0 1];

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

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);