Quickguide LLS Problem

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

The linear least squares (lls) problem is defined as

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}||Cx-d||\\&\\s/t&{\begin{array}{lcccl}x_{L}&\leq &x&\leq &x_{U},\\b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}\end{array}}$

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $d\in \mathbb {R} ^{M}$, $C\in \mathbb {R} ^{M\times n}$, $A\in \mathbb {R} ^{m_{1}\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$.

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);