Evolution and Optimum Seeking

62 Hill climbing Strategies
inequality constraints. The starting point of the search does not need to lie in the feasible
region. For this case Box suggests locating an allowed point by minimizing the function
with
until
~F(x) =;
j(x) =
mX
j=1
Gj(x) j(x)
( 0 if Gj(x) 0
1 otherwise
~F(x) =0
(3.23)
The two most important di erences from the Nelder-Mead strategy are the use of more
vertices and the expansion of the polyhedron at each normal re ection. Both measures
are intended to prevent the complex from eventually spanning only a subspace of reduced
dimensionality, especially at active constraints. If an allowed starting point isgiven or
has been found, it de nes one of the n +1 N 2 n vertices of the polyhedron. The
remaining vertex points are xed by a random process in which each vector inside the
closed region de ned by the explicit constraints has an equal probability of selection. If an
implicit constraint is violated, the new point is displaced stepwise towards the midpoint
of the allowed vertices that have already been de ned until it satis es all the constraints.
Implicit constraints Gj(x) 0 are dealt with similarly during the course of the minimum
search. If an explicit boundary is crossed, xi
back in the allowed region to a value near the boundary.
The details of the algorithm are as follows:
ai, the o ending variable is simply set
Step 0: (Initialization)
Choose a starting point x (0) andanumber of vertices N n +1 (e.g.,
N =2n). Number the constraints such that the rst j m 1 each depend
only on one variable, x`j (Gj(x`j), explicit form).
Test whether x (0) satis es all the constraints.
If not, then construct a substitute objective function according to Equation
(3.23).
Set up the initial complex as follows:
x (01) = x (0)
and x (0 ) = x (0) + nP
zi ei for = 2(1)N,
i=1
where the zi are uniformly distributed random numbers from the range
( [aibi], if constraints are given in the form ai xi bi
otherwise h x (0)
i
; 0:5 s x (0)
i
If Gj(x (0 ) ) < 0foranyj m1 > 1
(0 )
replace x `j 2 x (01) (0 )
`j ; x `j :
+0:5 s] where, e.g., s =1:
If Gj(x (0 ) ) < 0foranyj m > 1
replace x (0 ) 0:5[x (0 ) + 1 P;1
x ;1
=1
(0 ) ].