## 12Constrained Global Optimization

In glc_prob there are 30 global mixed-integer nonlinear programming test problems with sizes to 20 variables and 5 constrains. In order to define the problem n and solve it execute the following in Matlab:
Prob   = probInit('glc_prob',n);
Result = tomRun('',Prob);
The basic structure of a constrained global optimization problems is the following

 min x
f(x)

s/t
 −∞ < xL ≤ x ≤ xU < ∞ bL ≤ A x ≤ bU cL ≤ c(x) ≤ cU, xj N    j I,
(14)

where x, xL, xU Rn, f(x) R, A Rm1 × n, bL,bU Rm1 and cL,c(x),cU Rm2. The variables x I, the index subset of 1,...,n, are restricted to be integers.

The following files are required to define a problem of this category in TOMLAB.

File: tomlab/quickguide/glcQG_f.m, glcQG_c.m
f:   Function
c:   Constraints
An example of a problem of this class, (that is also found in the TOMLAB quickguide) is glcQG:

File: tomlab/quickguide/glcQG.m
% glcQG is a small example problem for defining and solving
% constrained global programming problems using the TOMLAB format.

Name = 'Hock-Schittkowski 59';
u = [75.196    3.8112    0.0020567  1.0345E-5  6.8306    0.030234   1.28134E-3 ...
2.266E-7  0.25645   0.0034604  1.3514E-5  28.106    5.2375E-6  6.3E-8     ...
7E-10     3.405E-4  1.6638E-6  2.8673     3.5256E-5];

x_L = [0 0]';     % Lower bounds for x.
x_U = [75 65]';   % Upper bounds for x.
b_L = []; b_U = []; A = []; % Linear constraints
c_L = [0 0 0];    % Lower bounds for nonlinear constraints.
c_U = [];         % Upper bounds for nonlinear constraints.
x_opt = [13.55010424 51.66018129]; % Optimum vector
f_opt = -7.804226324;              % Optimum
x_min = x_L;      % For plotting
x_max = x_U;      % For plotting
x_0 = [90 10]';   % If running local solver

Prob = glcAssign('glcQG_f', x_L, x_U, Name, A, b_L, b_U, ...
'glcQG_c', c_L, c_U, x_0, ...
[], [], [], [], ...
[], x_min, x_max, f_opt, x_opt);

Prob.user.u = u;
Prob.optParam.MaxFunc = 1500;

Result = tomRun('glcFast', Prob, 1);
%Result = tomRun('glcSolve', Prob, 1);
%Result = tomRun('lgo', Prob, 1);
%Result = tomRun('oqnlp', Prob, 1);