Mercurial > matrix-functions
view matrixcomp/mdsmax.m @ 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Wed, 06 May 2015 14:56:53 +0200 |
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function [x, fmax, nf] = mdsmax(fun, x, stopit, savit, varargin) %MDSMAX Multidirectional search method for direct search optimization. % [x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to % maximize the function FUN, using the starting vector x0. % The method of multidirectional search is used. % Output arguments: % x = vector yielding largest function value found, % fmax = function value at x, % nf = number of function evaluations. % The iteration is terminated when either % - the relative size of the simplex is <= STOPIT(1) % (default 1e-3), % - STOPIT(2) function evaluations have been performed % (default inf, i.e., no limit), or % - a function value equals or exceeds STOPIT(3) % (default inf, i.e., no test on function values). % The form of the initial simplex is determined by STOPIT(4): % STOPIT(4) = 0: regular simplex (sides of equal length, the default), % STOPIT(4) = 1: right-angled simplex. % Progress of the iteration is not shown if STOPIT(5) = 0 (default 1). % If a non-empty fourth parameter string SAVIT is present, then % `SAVE SAVIT x fmax nf' is executed after each inner iteration. % NB: x0 can be a matrix. In the output argument, in SAVIT saves, % and in function calls, x has the same shape as x0. % MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional % arguments to be passed to fun, via feval(fun,x,P1,P2,...). % This implementation uses 2n^2 elements of storage (two simplices), where x0 % is an n-vector. It is based on the algorithm statement in [2, sec.3], % modified so as to halve the storage (with a slight loss in readability). % References: % [1] V. J. Torczon, Multi-directional search: A direct search algorithm for % parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989. % [2] V. J. Torczon, On the convergence of the multidirectional search % algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145. % [3] N. J. Higham, Optimization by direct search in matrix computations, % SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. % [4] N. J. Higham, Accuracy and Stability of Numerical Algorithms, % Second edition, Society for Industrial and Applied Mathematics, % Philadelphia, PA, 2002; sec. 20.5. x0 = x(:); % Work with column vector internally. n = length(x0); mu = 2; % Expansion factor. theta = 0.5; % Contraction factor. % Set up convergence parameters etc. if nargin < 3 | isempty(stopit), stopit(1) = 1e-3; end tol = stopit(1); % Tolerance for cgce test based on relative size of simplex. if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations. if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values. if length(stopit) == 3, stopit(4) = 0; end % Default initial simplex. if length(stopit) == 4, stopit(5) = 1; end % Default: show progress. trace = stopit(5); if nargin < 4, savit = []; end % File name for snapshots. V = [zeros(n,1) eye(n)]; T = V; f = zeros(n+1,1); ft = f; V(:,1) = x0; f(1) = feval(fun,x,varargin{:}); fmax_old = f(1); if trace, fprintf('f(x0) = %9.4e\n', f(1)), end k = 0; m = 0; % Set up initial simplex. scale = max(norm(x0,inf),1); if stopit(4) == 0 % Regular simplex - all edges have same length. % Generated from construction given in reference [18, pp. 80-81] of [1]. alpha = scale / (n*sqrt(2)) * [ sqrt(n+1)-1+n sqrt(n+1)-1 ]; V(:,2:n+1) = (x0 + alpha(2)*ones(n,1)) * ones(1,n); for j=2:n+1 V(j-1,j) = x0(j-1) + alpha(1); x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); end else % Right-angled simplex based on co-ordinate axes. alpha = scale*ones(n+1,1); for j=2:n+1 V(:,j) = x0 + alpha(j)*V(:,j); x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); end end nf = n+1; size = 0; % Integer that keeps track of expansions/contractions. flag_break = 0; % Flag which becomes true when ready to quit outer loop. while 1 %%%%%% Outer loop. k = k+1; % Find a new best vertex x and function value fmax = f(x). [fmax,j] = max(f); V(:,[1 j]) = V(:,[j 1]); v1 = V(:,1); if ~isempty(savit), x(:) = v1; eval(['save ' savit ' x fmax nf']), end f([1 j]) = f([j 1]); if trace fprintf('Iter. %2.0f, inner = %2.0f, size = %2.0f, ', k, m, size) fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ... 100*(fmax-fmax_old)/(abs(fmax_old)+eps)) end fmax_old = fmax; % Stopping Test 1 - f reached target value? if fmax >= stopit(3) msg = ['Exceeded target...quitting\n']; break % Quit. end m = 0; while 1 %%% Inner repeat loop. m = m+1; % Stopping Test 2 - too many f-evals? if nf >= stopit(2) msg = ['Max no. of function evaluations exceeded...quitting\n']; flag_break = 1; break % Quit. end % Stopping Test 3 - converged? This is test (4.3) in [1]. size_simplex = norm(V(:,2:n+1)- v1(:,ones(1,n)),1) / max(1, norm(v1,1)); if size_simplex <= tol msg = sprintf('Simplex size %9.4e <= %9.4e...quitting\n', ... size_simplex, tol); flag_break = 1; break % Quit. end for j=2:n+1 % ---Rotation (reflection) step. T(:,j) = 2*v1 - V(:,j); x(:) = T(:,j); ft(j) = feval(fun,x,varargin{:}); end nf = nf + n; replaced = ( max(ft(2:n+1)) > fmax ); if replaced for j=2:n+1 % ---Expansion step. V(:,j) = (1-mu)*v1 + mu*T(:,j); x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); end nf = nf + n; % Accept expansion or rotation? if max(ft(2:n+1)) > max(f(2:n+1)) V(:,2:n+1) = T(:,2:n+1); f(2:n+1) = ft(2:n+1); % Accept rotation. else size = size + 1; % Accept expansion (f and V already set). end else for j=2:n+1 % ---Contraction step. V(:,j) = (1+theta)*v1 - theta*T(:,j); x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); end nf = nf + n; replaced = ( max(f(2:n+1)) > fmax ); % Accept contraction (f and V already set). size = size - 1; end if replaced, break, end if trace & rem(m,10) == 0, fprintf(' ...inner = %2.0f...\n',m), end end %%% Of inner repeat loop. if flag_break, break, end end %%%%%% Of outer loop. % Finished. if trace, fprintf(msg), end x(:) = v1;