Mixed-Integer Quadratic with QCP Constraints

6 Mixed-Integer Quadratic w QC

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:

min

f(x) =

x^T F x + c^T x

s/t

x_L

≤

x_U

b_L

≤

b_U

x^T Q⁽ⁱ⁾ x + a^(i)T x

≤

r_U⁽ⁱ⁾,

i=1,…,n_qc

x_i integer

(7)

where c, x, x_L, x_U, a⁽ⁱ⁾

Rⁿ, F, Q⁽ⁱ⁾

R^{n× n}, A

R^m×
n and b_L,b_U

R^m. r_U⁽ⁱ⁾ is a scalar. The variables x

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);

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