Quickguide L1LIN Problem

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

The linearly constrained L1LIN (L1LIN) problem is defined as

${\begin{array}{cccccc}\min \limits _{x}&{5}{l}{|Cx-y)|+alpha*|Lx|}\\{\mbox{subject to}}&x_{L}&\leq &x&\leq &x_{U}\\{}&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $b_{L},b_{U}\in \mathbb {R} ^{m_{2}}$, $A\in \mathbb {R} ^{m_{1}\times n}$, $C\in \mathbb {R} ^{m_{2}\times n}$, $y\in \mathbb {R} ^{m_{2}}$, $L\in \mathbb {R} ^{n\times b}$ and $\alpha \in \mathbb {R} ^{1}$.

The L1Lin solution can be obtained by the use of any suitable linear TOMLAB solver.

The following file illustrates how to solve an L1Lin problem in TOMLAB.

File: tomlab/quickguide/L1LinQG.m

Open the file for viewing, and execute L1LinQG in Matlab.

 % L1LinQG is a small example problem for defining and solving
 % a linearly constrained linear L1 problem using the TOMLAB format.
 
 Name='L1LinSolve test example';      % Problem name, not required.
 n = 6;                              
 x_L = -10*ones(n,1);                 % Lower bounds on x
 x_U =  10*ones(n,1);                 % Upper bounds on x
 x_0 = (x_L + x_U) / 2;               % Starting point
 
 C = spdiags([1 2 3 4 5 6]', 0, n, n); % C matrix
 y = 1.5*ones(n,1);                    % Data vector
 
 % Matrix defining linear constraints
 A = [1 1 0 0 0 0];
 b_L = 1;                  % Lower bounds on the linear inequalities
 b_U = 1;                  % Upper bounds on the linear inequalities
 
 % Defining damping matrix
 Prob.LS.damp = 1;
 Prob.LS.L = spdiags(ones(6,1)*0.01, 0, 6, 6);
 
 % See 'help llsAssign' for more information.
 Prob = llsAssign(C, y, x_L, x_U, Name, x_0, ...
                             [], [], [], ...
                             A, b_L, b_U);
                         
 Prob.SolverL1 = 'lpSimplex';
 Result = tomRun('L1LinSolve', Prob, 1);
 % Prob.SolverL1 = 'MINOS';
 % Result = tomRun('L1LinSolve', Prob, 1);
 % Prob.SolverL1 = 'CPLEX';
 % Result = tomRun('L1LinSolve', Prob, 1);