Evolution and Optimum Seeking

58 Hill climbing Strategies
of factorial design (see for example Davies, 1954). The minimum number according to
Brooks and Mickey (1961) is n +1. Thus instead of a single starting point, n +1vertices
are used. They are arranged so as to be equidistant from each other: for n = 2 in an
equilateral triangle for n =3atetrahedron and in general a polyhedron, also referred to
as a simplex. The objective function is evaluated at all the vertices. The iteration rule is:
Replace the vertex with the largest objective function value by a new one situated at its
re ection in the midpoint of the other n vertices. This rule aims to locate the new point
at an especially promising place. If one lands near a minimum, the newest vertex can
also be the worst. In this case the second worst vertex should be re ected. If the edge
length of the polyhedron is not changed, the search eventually stagnates. The polyhedra
rotate about the vertex with the best objective function value. A closer approximation to
the optimum can only be achieved by halving the edge lengths of the simplex. Spendley,
Hext, and Himsworth suggest doing this whenever a vertex is common to more than
1:65 n +0:05 n 2 consecutive polyhedra. Himsworth (1962) holds that this strategy is
especially advantageous when the number of variables is large and the determination of
the objective function prone to error.
To this basic procedure, various modi cations have been proposed by, among others,
Nelder and Mead (1965), Box (1965), Ward, Nag, and Dixon (1969), and Dambrauskas
(1970, 1972). Richardson and Kuester (1973) have provided a complete program. The
most common version is that of Nelder and Mead, in which the main di erence from the
basic procedure is that the size and shape of the simplex is modi ed during the run to
suit the conditions at each stage.
The algorithm, with an extension by O'Neill (1971), runs as follows:
Step 0: (Initialization)
Choose a starting point x (00) , initial step lengths s (0)
i for all i = 1(1)n
(if no better scaling is known, s (0)
i = 1), and an accuracy parameter " > 0
(e.g., " =10 ;8 ). Set c =1andk =0.
Step 1: (Establish the initial simplex)
x (k ) = x (k0) + cs (0) e for all = 1(1)n.
Step 2: (Determine worst and best points for the normal re ection)
Determine the indices w (worst point) and b (best point) such that
F (x (kw) ) = maxfF (x (k ) ) = 0(1)ng
F (x (kb) ) = minfF (x (k ) ) = 0(1)ng
Construct x = 1
n
nP
=0 6=w
If F (x 0 ) <
>:
> 1 set x (k+1w) = x 0 and go to step 8
=1 go to step 5
=0 go to step 6: