Quickguide MINLP Problem

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

The mixed-integer nonlinear programming (minlp) problem is defined as

${\begin{array}{ll}\min \limits _{x}&f(x)\\&\\s/t&{\begin{array}{llcccll}-\infty <&x_{L}&\leq &x&\leq &x_{U}&<\infty \\&b_{L}&\leq &Ax&\leq &b_{U}&\\&c_{L}&\leq &c(x)&\leq &c_{U},&~x_{j}\in \mathbb {N} \ ~~\forall j\in \$I\$,\\\end{array}}\end{array}}$

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $f(x)\in \mathbb {R}$, $A\in \mathbb {R} ^{m_{1}\times n}$, $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$ and $c_{L},c(x),c_{U}\in \mathbb {R} ^{m_{2}}$. The variables $x\in I$, the index subset of $1,...,n$, are restricted to be integers.

The following files define a problem 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.

The following file illustrates how to solve a MINLP problem in TOMLAB. Also view the m-files specified above for more information.

File: tomlab/quickguide/minlpQG.m

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

 % minlpQG is a small example problem for defining and solving
 % mixed-integer nonlinear programming problems using the TOMLAB format.
 
 Name='minlp1Demo - Kocis/Grossman.';
 
 IntVars   = logical([ 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);