## 11LLS Problem

The linear least squares (lls) problem is defined as
 min x
f(x) =
1
2
|| C xd ||

s/t
 xL ≤ x ≤ xU, bL ≤ A x ≤ bU
(13)
where x, xL, xU Rn, d RM, C RM × n, A Rm1 × n and bL,bU Rm1.

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.