# The Matrix Computation Toolbox

### Nick Higham (view profile)

11 Sep 2002 (Updated )

A collection of M-files for carrying out various numerical linear algebra tasks.

nmsmax(fun, x, stopit, savit, varargin)
```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);
```