Quickguide LLS Problem
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This page is part of the Quickguide Manual. See Quickguide. |
The linear least squares (lls) problem is defined as
where Failed to parse (unknown function "\MATHSET"): {\displaystyle x, x_L, x_U \in \MATHSET{R}^n}
, Failed to parse (unknown function "\MATHSET"): {\displaystyle d \in \MATHSET{R}^M}
, Failed to parse (unknown function "\MATHSET"): {\displaystyle C \in \MATHSET{R}^{M \times n}}
, Failed to parse (unknown function "\MATHSET"): {\displaystyle A \in \MATHSET{R}^{m_1 \times n}}
and
Failed to parse (unknown function "\MATHSET"): {\displaystyle b_L,b_U \in \MATHSET{R}^{m_1}}
.
The following file defines and solves a problem in TOMLAB.
File: tomlab/quickguide/llsQG.m
Open the file for viewing, and execute llsQG in Matlab.
% llsQG is a small example problem for defining and solving
% linear least squares using the TOMLAB format.
Name='LSSOL test example'; % Problem name, not required.
n = 9;
x_L = [-2 -2 -inf, -2*ones(1,6)]'; % Lower bounds on x
x_U = 2*ones(9,1); % Upper bounds on x
% Matrix defining linear constraints
A = [ ones(1,8) 4; 1:4,-2,1 1 1 1; 1 -1 1 -1, ones(1,5)];
b_L = [2 -inf -4]'; % Lower bounds on the linear inequalities
b_U = [inf -2 -2]'; % Upper bounds on the linear inequalities
% Vector m x 1 with observations in objective ||Cx -y(t)||
y = 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; 1 1 3 1 1 1 -1 -1 -3; ...
1 1 1 4 1 1 1 1 1;1 1 1 3 1 1 1 1 1;1 1 2 1 1 0 0 0 -1; ...
1 1 1 1 0 1 1 1 1;1 1 1 0 1 1 1 1 1;1 1 0 1 1 1 2 2 3; ...
1 0 1 1 1 1 0 2 2];
% 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);
% x_opt estimate.
x_opt = [2 1.57195927 -1.44540327 -0.03700275 0.54668583 0.17512363 ...
-1.65670447 -0.39474418 0.31002899];
f_opt = 0.1390587318; % Estimated optimum.
% See 'help llsAssign' for more information.
Prob = llsAssign(C, y, x_L, x_U, Name, x_0, ...
[], [], [], ...
A, b_L, b_U, ...
x_min, x_max, f_opt, x_opt);
Result = tomRun('clsSolve', Prob, 1);
%Result = tomRun('nlssol', Prob, 1);
%Result = tomRun('snopt', Prob, 1);
%Result = tomRun('lssol', Prob, 1);