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8  Mixed-Integer Nonlinear Programming

In minlp_prob there are 14 mixed-integer nonlinear programming test problems with sizes to nearly 50 variables and nearly 50 constraints. In order to define problem number n and solve it execute the following in Matlab:

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

The basic structure of a general mixed-integer nonlinear programming problem is the following



where x, x_L, x_U Rⁿ, f(x) R, A R^{m₁ × n}, b_L,b_U R^m₁ and c_L,c(x),c_U R^m₂. 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/minlpQG_f.m, minlpQG_g.m, minlpQG_H.m, minlpQG_c.m, minlpQG_dc.m, minlpQG_d2c.m

  f:   Function value
  g:   Gradient vector
  H:   Hessian matrix
  c:   Nonlinear constraint vector
  dc:  Nonlinear constraint gradient matrix
  d2c: The second part of the Hessian to the Lagrangian
       function for the nonlinear constraints.

An example of a problem of this class, (that is also found in the TOMLAB Quickguide) is minlpQG

File: tomlab/quickguide/minlpQG.m

% minlpQG is a small example problem for defining and solving
% mixed-integer nonlinear programming problems using the TOMLAB format.

Name='minlp1Demo - Kocis/Grossman.';

IntVars   = [ 0 0 1 1 1 ]; % Integer variables: x(3)-x(5)
VarWeight = [ ];           % No priorities given

% There are divisions and square roots involving x(2), so we must
% have a small but positive value for the lower bound on x(2).
BIG = 1E8;

x_L = [ 0   1/BIG   0 0 0 ]';    % Lower bounds on x
x_U = [ BIG  BIG    1 1 1 ]';   % Upper bounds on x

% Three linear constraints
A = [1 0     1  0 0 ; ...
     0 1.333 0  1 0 ; ...
     0 0    -1 -1 1 ];

b_L = [];               % No lower bounds
b_U = [1.6 ; 3 ; 0];    % Upper bounds

c_L = [1.25;3];         % Two nonlinear constraints
c_U = c_L;              % c_L==c_U implies equality

x_0 = ones(5,1);        % Initial value

x_opt = [1.12,1.31,0,1,1]';  % One optimum known
f_opt = 7.6672;              % Value f(x_opt)

x_min = [-1 -1 0 0 0];  % Used for plotting, lower bounds
x_max = [ 1  1 1 1 1]; % Used for plotting, upper bounds

HessPattern = spalloc(5,5,0);  % All elements in Hessian are zero.
ConsPattern = [ 1 0 1 0 0; ...  % Sparsity pattern of nonlinear
         0 1 0 1 0 ];    % constraint gradient

fIP = [];        % An upper bound on the IP value wanted. Makes it possible
xIP = [];        % to cut branches. xIP: the x value giving fIP

% Generate the problem structure using the TOMLAB Quick format
Prob = minlpAssign('minlpQG_f', 'minlpQG_g', 'minlpQG_H', HessPattern, ...
                  x_L, x_U, Name, x_0, ...
                  IntVars, VarWeight, fIP, xIP, ...
                  A, b_L, b_U, 'minlpQG_c', 'minlpQG_dc', 'minlpQG_d2c', ...
            ConsPattern, c_L, c_U, ...
                  x_min, x_max, f_opt, x_opt);
Prob.DUNDEE.optPar(20) = 1;
Prob.P = 1;   % Needed in minlpQG_xxx files

% Get default TOMLAB solver for your current license, for "minlp" problems
% Solver = GetSolver('minlp');

% Call driver routine tomRun, 3rd argument > 0 implies call to PrintResult

Result  = tomRun('minlpBB',Prob,2);

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