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Create local repository for matrix toolboxes. Step #0 done.
author Antonio Pino Robles <data.script93@gmail.com>
date Wed, 06 May 2015 14:56:53 +0200
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1 function [x, fmax, nf] = nmsmax(fun, x, stopit, savit, varargin)
2 %NMSMAX Nelder-Mead simplex method for direct search optimization.
3 % [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to
4 % maximize the function FUN, using the starting vector x0.
5 % The Nelder-Mead direct search method is used.
6 % Output arguments:
7 % x = vector yielding largest function value found,
8 % fmax = function value at x,
9 % nf = number of function evaluations.
10 % The iteration is terminated when either
11 % - the relative size of the simplex is <= STOPIT(1)
12 % (default 1e-3),
13 % - STOPIT(2) function evaluations have been performed
14 % (default inf, i.e., no limit), or
15 % - a function value equals or exceeds STOPIT(3)
16 % (default inf, i.e., no test on function values).
17 % The form of the initial simplex is determined by STOPIT(4):
18 % STOPIT(4) = 0: regular simplex (sides of equal length, the default)
19 % STOPIT(4) = 1: right-angled simplex.
20 % Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
21 % If a non-empty fourth parameter string SAVIT is present, then
22 % `SAVE SAVIT x fmax nf' is executed after each inner iteration.
23 % NB: x0 can be a matrix. In the output argument, in SAVIT saves,
24 % and in function calls, x has the same shape as x0.
25 % NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
26 % arguments to be passed to fun, via feval(fun,x,P1,P2,...).
27
28 % References:
29 % N. J. Higham, Optimization by direct search in matrix computations,
30 % SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
31 % C. T. Kelley, Iterative Methods for Optimization, Society for Industrial
32 % and Applied Mathematics, Philadelphia, PA, 1999.
33
34 x0 = x(:); % Work with column vector internally.
35 n = length(x0);
36
37 % Set up convergence parameters etc.
38 if nargin < 3 | isempty(stopit), stopit(1) = 1e-3; end
39 tol = stopit(1); % Tolerance for cgce test based on relative size of simplex.
40 if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations.
41 if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values.
42 if length(stopit) == 3, stopit(4) = 0; end % Default initial simplex.
43 if length(stopit) == 4, stopit(5) = 1; end % Default: show progress.
44 trace = stopit(5);
45 if nargin < 4, savit = []; end % File name for snapshots.
46
47 V = [zeros(n,1) eye(n)];
48 f = zeros(n+1,1);
49 V(:,1) = x0; f(1) = feval(fun,x,varargin{:});
50 fmax_old = f(1);
51
52 if trace, fprintf('f(x0) = %9.4e\n', f(1)), end
53
54 k = 0; m = 0;
55
56 % Set up initial simplex.
57 scale = max(norm(x0,inf),1);
58 if stopit(4) == 0
59 % Regular simplex - all edges have same length.
60 % Generated from construction given in reference [18, pp. 80-81] of [1].
61 alpha = scale / (n*sqrt(2)) * [ sqrt(n+1)-1+n sqrt(n+1)-1 ];
62 V(:,2:n+1) = (x0 + alpha(2)*ones(n,1)) * ones(1,n);
63 for j=2:n+1
64 V(j-1,j) = x0(j-1) + alpha(1);
65 x(:) = V(:,j); f(j) = feval(fun,x,varargin{:});
66 end
67 else
68 % Right-angled simplex based on co-ordinate axes.
69 alpha = scale*ones(n+1,1);
70 for j=2:n+1
71 V(:,j) = x0 + alpha(j)*V(:,j);
72 x(:) = V(:,j); f(j) = feval(fun,x,varargin{:});
73 end
74 end
75 nf = n+1;
76 how = 'initial ';
77
78 [temp,j] = sort(f);
79 j = j(n+1:-1:1);
80 f = f(j); V = V(:,j);
81
82 alpha = 1; beta = 1/2; gamma = 2;
83
84 while 1 %%%%%% Outer (and only) loop.
85 k = k+1;
86
87 fmax = f(1);
88 if fmax > fmax_old
89 if ~isempty(savit)
90 x(:) = V(:,1); eval(['save ' savit ' x fmax nf'])
91 end
92 if trace
93 fprintf('Iter. %2.0f,', k)
94 fprintf([' how = ' how ' ']);
95 fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ...
96 100*(fmax-fmax_old)/(abs(fmax_old)+eps))
97 end
98 end
99 fmax_old = fmax;
100
101 %%% Three stopping tests from MDSMAX.M
102
103 % Stopping Test 1 - f reached target value?
104 if fmax >= stopit(3)
105 msg = ['Exceeded target...quitting\n'];
106 break % Quit.
107 end
108
109 % Stopping Test 2 - too many f-evals?
110 if nf >= stopit(2)
111 msg = ['Max no. of function evaluations exceeded...quitting\n'];
112 break % Quit.
113 end
114
115 % Stopping Test 3 - converged? This is test (4.3) in [1].
116 v1 = V(:,1);
117 size_simplex = norm(V(:,2:n+1)-v1(:,ones(1,n)),1) / max(1, norm(v1,1));
118 if size_simplex <= tol
119 msg = sprintf('Simplex size %9.4e <= %9.4e...quitting\n', ...
120 size_simplex, tol);
121 break % Quit.
122 end
123
124 % One step of the Nelder-Mead simplex algorithm
125 % NJH: Altered function calls and changed CNT to NF.
126 % Changed each `fr < f(1)' type test to `>' for maximization
127 % and re-ordered function values after sort.
128
129 vbar = (sum(V(:,1:n)')/n)'; % Mean value
130 vr = (1 + alpha)*vbar - alpha*V(:,n+1); x(:) = vr; fr = feval(fun,x,varargin{:});
131 nf = nf + 1;
132 vk = vr; fk = fr; how = 'reflect, ';
133 if fr > f(n)
134 if fr > f(1)
135 ve = gamma*vr + (1-gamma)*vbar; x(:) = ve; fe = feval(fun,x,varargin{:});
136 nf = nf + 1;
137 if fe > f(1)
138 vk = ve; fk = fe;
139 how = 'expand, ';
140 end
141 end
142 else
143 vt = V(:,n+1); ft = f(n+1);
144 if fr > ft
145 vt = vr; ft = fr;
146 end
147 vc = beta*vt + (1-beta)*vbar; x(:) = vc; fc = feval(fun,x,varargin{:});
148 nf = nf + 1;
149 if fc > f(n)
150 vk = vc; fk = fc;
151 how = 'contract,';
152 else
153 for j = 2:n
154 V(:,j) = (V(:,1) + V(:,j))/2;
155 x(:) = V(:,j); f(j) = feval(fun,x,varargin{:});
156 end
157 nf = nf + n-1;
158 vk = (V(:,1) + V(:,n+1))/2; x(:) = vk; fk = feval(fun,x,varargin{:});
159 nf = nf + 1;
160 how = 'shrink, ';
161 end
162 end
163 V(:,n+1) = vk;
164 f(n+1) = fk;
165 [temp,j] = sort(f);
166 j = j(n+1:-1:1);
167 f = f(j); V = V(:,j);
168
169 end %%%%%% End of outer (and only) loop.
170
171 % Finished.
172 if trace, fprintf(msg), end
173 x(:) = V(:,1);