Quickguide L1 Problem

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

The constrained L1 (L1) problem is defined as

${\begin{array}{cccccc}\min \limits _{x}&{\sum _{i}|r_{i}(x)|}\\{\mbox{subject to}}&x_{L}&\leq &x&\leq &x_{U}\\{}&b_{L}&\leq &Ax&\leq &b_{U}\\{}&c_{L}&\leq &c(x)&\leq &c_{U}\\\end{array}}$

where $x,x_{L},x_{U}\in R^{n}$, $r(x)\in R^{N}$, $c(x),c_{L},c_{U}\in R^{m_{1}}$, $b_{L},b_{U}\in R^{m_{2}}$ and $A\in R^{m_{2}\times n}$.

The L1 solution can be obtained by the use of any suitable nonlinear TOMLAB solver.

The following files define a problem in TOMLAB.

File: tomlab/quickguide/L1QG_r.m, L1QG_J.m

r: Residual vector
J: Jacobian matrix

The following file illustrates how to solve an L1 problem in TOMLAB. Also view the m-files specified above for more information.

File: tomlab/quickguide/L1QG.m

Open the file for viewing, and execute L1QG in Matlab.

 % L1QG is a small example problem for defining and solving
 % an L1 problem using the TOMLAB format.
 
 Name = 'Madsen-Tinglett 1';
 x_0  = [1;1];        % Initial value
 x_L  = [-10;-10];    % Lower bounds on x
 x_U  = [10;10];      % Upper bounds on x
 
 % Solve the problem min max |r_i(x)|, where i = 1,2,3
 % Solve the problem by eliminating abs, doubling the residuals, reverse sign
 % i.e. min max [r_1; r_2; r_3; -r_1; -r_2; -r_3];
 y = [-1.5; -2.25; -2.625]; % The data values
 t = [];                 % No time vector used
 
 % Generate the problem structure using the Tomlab format
 % as a standard least squares problem.
 %
 % The part in the residuals dependent on x are defined in L1ex_r.m 
 % The Jacobian is defined in L1ex_J.m
 
 Prob = clsAssign('L1QG_r', 'L1QG_J', [], x_L, x_U, Name, x_0, y, t);
 
 % Set JacPattern, L1Solve will compute correct ConsPattern
 Prob.JacPattern  = ones(length(y),2);
 
 % Set the optimal values into the structure (for nice result presenting)
 
 Prob.x_opt=[3, 0.5];
 Prob.f_opt=0;
 
 % Use the standard method in L1Solve
 Prob.L1Type = 1;
 
 % Get the default solver
 Solver = GetSolver('con',1,0);
 
 Prob.SolverL1 = Solver;
 
 % One may set the solver directly:
 %Prob.SolverL1 = 'snopt';
 %Prob.SolverL1 = 'npsol';
 %Prob.SolverL1 = 'minos';
 %Prob.SolverL1 = 'conSolve';
 
 % These statements generate ASCII log files when running SNOPT
 if strcmpi(Solver,'snopt') 
    Prob.SOL.PrintFile = 'snopt.txt';
    Prob.SOL.SummFile  = 'snopts.txt';
    Prob.SOL.optPar(1) = 10;   % Print level in SNOPT
    %Prob.SOL.optPar(13) = 3;   % Verify level in SNOPT
 end
 
 % Set print level 2 to get output from PrintResult at the end
 PriLev = 2;
 Prob.PriLevOpt = 0;
 
 Result  = tomRun('L1Solve', Prob, PriLev);