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12  MILLS Problem

The mixed-integer linear least squares (mills) problem is defined as where x, x_L, x_U Rⁿ, d R^M, C R^{M × n}, A R^{m₁ × n} and b_L,b_U R^m₁. The variables x I, the index subset of 1,...,n, are restricted to be integers.

The following file defines and solves a problem in TOMLAB.

File: tomlab/quickguide/millsQG.m

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

% millsQG is a small example problem for defining and solving
% mixed-integr linear least squares using the TOMLAB format.
Name='MILLS test example';           % Problem name, not required.
n = 10;
x_L = zeros(n,1);                    % Lower bounds on x
x_U = 20*ones(n,1);                  % Upper bounds on x

% Matrix defining linear constraints
A   = [ ones(1,n) ; 1:10; -2 1 -1 1 -1 1 -1 1 -1 1];
b_L = [35  -inf  29]';    % Lower bounds on the linear inequalities
b_U = [inf  120 210]';    % Upper bounds on the linear inequalities

% Vector m x 1 with observations in objective ||Cx -y(t)||
y = 2.5*ones(10,1);

% Matrix m x n in objective ||Cx -y(t)||
C = [ ones(1,n); 1 2 1 1 1 1 2 0 0 0; 1 1 3 1 1 1 -1 -1 -3 1; ...
   1 1 1 4 1 1 1 1 1 0;1 1 1 3 1 1 1 1 1 0;1 1 2 1 1 0 0 0 -1 1; ...
   1 1 1 1 0 1 1 1 1 0;1 1 1 0 1 1 1 1 1 1;1 1 0 1 1 1 2 2 3 0; ...
   1 0 1 1 1 1 0 2 2 1];

% Starting point.
x_0 = 1./[1:n]';

% x_min and x_max are only needed if doing plots.
x_min = -ones(n,1);
x_max =  ones(n,1);

% See 'help llsAssign' for more information.
Prob = llsAssign(C, y, x_L, x_U, Name, x_0, ...
   [], [], [], ...
   A, b_L, b_U);

IntVars = [ones(n-2,1);zeros(2,1)];

Prob = lls2qp(Prob, IntVars);

Result = tomRun('cplex', Prob, 1);
% Result = tomRun('minlpBB', Prob, 1);

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