Evolution and Optimum Seeking

64 Hill climbing Strategies
Step 9: (Termination)
Determine the index b (best vertex) such that
F (x (kb) )=minfF (x (k ) ) = 1(1)Ng:
End the search with the result x (kb) and F (x (kb) ):
Box himself reports that in numerical tests his complex strategy gives similar results
to the simplex method of Nelder and Mead, but both are inferior to the method of
Rosenbrock with regard to the number of objective function calls. He actually uses his
own modi cation of the Rosenbrock method. Investigation of the e ect of the number of
vertices of the complex and the expansion factor (in this case 2 n and 1.3 respectively)
lead him to the conclusion that neither value has a signi cant e ect on the e ciency of
the strategy. For n>5 he considers that a number of vertices N =2n is unnecessarily
high, especially when there are no constraints.
The convergence criterion appears very reliable. While Nelder and Mead require that
the standard deviation of all objective function values at the polyhedron vertices, referred
to its midpoint, must be less than a prescribed size, the complex search is only ended
when several consecutive values of the objective function are the same to computational
accuracy.
Because of the larger number of polyhedron vertices the complex method needs even
more storage space than the simplex strategy. The order of magnitude, O(n 2 ), remains
the same. No investigations are known of the computational e ort in the case of many
variables. Modi cations of the strategy are due to Guin (1968), Mitchell and Kaplan
(1968), and Dambrauskas (1970, 1972). Guin de nes a contraction rule with which an
allowed point can be generated even if the allowed region is not convex. This is not always
the case in the original method because the midpointtowhichtheworst vertex is re ected
is not tested for feasibility.
Mitchell nds that the initial con guration of the complex in uences the results obtained.
It is therefore better to place the vertices in a deterministic way rather than to
make a random choice. Dambrauskas combines the complex method with the step length
rule of the stochastic approximation. He requires that the step lengths or edge lengths of
the polyhedron go to zero in the limit of an in nite number of iterations, while their sum
tends to in nity. This measure may well increase the reliability of convergence however,
it also increases the cost. Beveridge and Schechter (1970) describe how the iteration rules
must be changed if the variables can take only discrete values. A practical application,
in which a process has to be optimized dynamically, is described by Tazaki, Shindo, and
Umeda (1970) this is the original problem for which Spendley, Hext, and Himsworth
(1962) conceived their simplex EVOP (evolutionary operation) procedure.
Compared to other numerical optimization procedures the polyhedra strategies have
the disadvantage that in the closing phase, near the optimum, they converge rather slowly
and sometimes even stagnate. The direction of progress selected by the re ection then
no longer coincides at all with the gradient direction. To remove this di culty it has
been suggested that information about the topology of the objective function, as given by
function values at the vertices of the polyhedron, be exploited to carry out a quadratic
interpolation. Such surface tting is familiar from the related methods of test planning and