WLS 加权最小二乘

% 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;