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.

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