www.gusucode.com > 矩阵计算工具箱 - mctoolbox源码程序 > matrixcomp\nmsmax.m
function [x, fmax, nf] = nmsmax(fun, x, stopit, savit, varargin) %NMSMAX Nelder-Mead simplex method for direct search optimization. % [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to % maximize the function FUN, using the starting vector x0. % The Nelder-Mead direct search method 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. % NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional % arguments to be passed to fun, via feval(fun,x,P1,P2,...). % References: % N. J. Higham, Optimization by direct search in matrix computations, % SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. % C. T. Kelley, Iterative Methods for Optimization, Society for Industrial % and Applied Mathematics, Philadelphia, PA, 1999. x0 = x(:); % Work with column vector internally. n = length(x0); % 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)]; f = zeros(n+1,1); 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; how = 'initial '; [temp,j] = sort(f); j = j(n+1:-1:1); f = f(j); V = V(:,j); alpha = 1; beta = 1/2; gamma = 2; while 1 %%%%%% Outer (and only) loop. k = k+1; fmax = f(1); if fmax > fmax_old if ~isempty(savit) x(:) = V(:,1); eval(['save ' savit ' x fmax nf']) end if trace fprintf('Iter. %2.0f,', k) fprintf([' how = ' how ' ']); fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ... 100*(fmax-fmax_old)/(abs(fmax_old)+eps)) end end fmax_old = fmax; %%% Three stopping tests from MDSMAX.M % Stopping Test 1 - f reached target value? if fmax >= stopit(3) msg = ['Exceeded target...quitting\n']; break % Quit. end % Stopping Test 2 - too many f-evals? if nf >= stopit(2) msg = ['Max no. of function evaluations exceeded...quitting\n']; break % Quit. end % Stopping Test 3 - converged? This is test (4.3) in [1]. v1 = V(:,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); break % Quit. end % One step of the Nelder-Mead simplex algorithm % NJH: Altered function calls and changed CNT to NF. % Changed each `fr < f(1)' type test to `>' for maximization % and re-ordered function values after sort. vbar = (sum(V(:,1:n)')/n)'; % Mean value vr = (1 + alpha)*vbar - alpha*V(:,n+1); x(:) = vr; fr = feval(fun,x,varargin{:}); nf = nf + 1; vk = vr; fk = fr; how = 'reflect, '; if fr > f(n) if fr > f(1) ve = gamma*vr + (1-gamma)*vbar; x(:) = ve; fe = feval(fun,x,varargin{:}); nf = nf + 1; if fe > f(1) vk = ve; fk = fe; how = 'expand, '; end end else vt = V(:,n+1); ft = f(n+1); if fr > ft vt = vr; ft = fr; end vc = beta*vt + (1-beta)*vbar; x(:) = vc; fc = feval(fun,x,varargin{:}); nf = nf + 1; if fc > f(n) vk = vc; fk = fc; how = 'contract,'; else for j = 2:n V(:,j) = (V(:,1) + V(:,j))/2; x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); end nf = nf + n-1; vk = (V(:,1) + V(:,n+1))/2; x(:) = vk; fk = feval(fun,x,varargin{:}); nf = nf + 1; how = 'shrink, '; end end V(:,n+1) = vk; f(n+1) = fk; [temp,j] = sort(f); j = j(n+1:-1:1); f = f(j); V = V(:,j); end %%%%%% End of outer (and only) loop. % Finished. if trace, fprintf(msg), end x(:) = V(:,1);