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<h3 class="section">1.8 A multidirectional search algorithm</h3>
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<h4 class="subheading">Helptext:</h4>
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<pre class="verbatim">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.
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