...

% function: fwlsmm.m
% purpose: FWLSMM finds a constrained minimum of a function of several variables.
%   FWLSMM attempts to solve problems of the form:
%    min beta*(W_ls*F(X)) + (1 - beta)*(max(W_mm*F(X)))
%     X
%
%    subject to:  A*X  <= B, Aeq*X  = Beq (linear constraints)
%                 C(X) <= 0, Ceq(X) = 0   (nonlinear constraints)
%                   LB <= X <= UB         (bounds)
%
% inputs: All inputs follows closely to FMINCON function, except for the
%         following 3 input parameters.
%   beta - weight to select between weighted least square or weighted min-max
%          optimisation. Must be between 0 - 1, inclusively. 0 for pure 
%          weighted min-max, 1 for pure weighted least square optimisation. 
%   wls  - weight for least squares optimisation. Must have same size as
%          the matrix returned by the cost function, F(X). Default value is
%          ones(R, C)/(R*C), where R and C are number of rows and columns of
%          the matrix returned by the cost function, F(X).
%   wmm  - weight for min-max optimisation. Must have same size as the
%          matrix returned by the cost function, F(X). Default value is
%          ones(R, C)/(R*C), where R and C are number of rows and columns of
%          the matrix returned by the cost function, F(X).
%
% outputs: All outputs follow closely to the one from FMINCON
%
% comment: As with the fmincon.m, this function also suffers from local
%   minima solution. Can try to compare the solution with the one obtained
%   from SOCP(sedumi). SOCP gives global minima
%
% author: LCC Corps
% date: Sept 12, 2008

function [x, fval, exitflag, output, lambda, grad, hessian] = ...
    fwlsmm(fun, x0, beta, wls, wmm, A, b, Aeq, beq, lb, ub, nonlcon, options)

% set parameters not imported to default value
if (nargin < 13),
    options = [];
    if (nargin < 12),
        nonlcon = [];
        if (nargin < 11),
            ub = [];
            if (nargin < 10),
                lb = [];
                if (nargin < 9),
                    beq = [];
                    if (nargin < 8),
                        Aeq = [];
                        if (nargin < 7),
                            b = [];
                            if (nargin < 6),
                                A = [];
                                if (nargin < 5),
                                    wmm = [];
                                    if (nargin < 4),
                                        wls = [];
                                        if (nargin < 3)
                                            beta = [];
                                        end;
                                    end;
                                end;
                            end;
                        end;
                    end;
                end;
            end;
        end;
    end;
end;
  
if (isempty(nonlcon)),
    nonlcon = @defnlcon;
end;

[rc, cc] = size(fun(x0));
if (isempty(wls)),
    wls = ones(rc, cc)/(rc*cc);
end;

if (isempty(wmm)),
    wmm = ones(rc, cc);
end;

% cost function having both weighted least squares and weighted min-max
cost = @(x) beta*sum(wls.*fun(x)) + (1-beta)*max(wmm.*fun(x));

% solving the weighted least squares and weighted min-max problem using
% fmincon
[x, fval, exitflag, output, lambda, grad, hessian] = ...
    fmincon(cost, x0, A, b, Aeq, beq, lb, ub, nonlcon, options);


% default non-linear contraints, in case the user did not pass in any
function [c, ceq] = defnlcon(x)
c = 0;
ceq = 0;