Table 1: Results of GPSA. The results were averaged over
100 independent trials where “SUCC %” indicates the ra-
tio of trials which succeed in attaining the global optimum
, “EVAL #” means the average number of function eval-
uations, and “VAR” means the variance of trials which
succeed. Functions (Yao et al., 1999; Schwefel, 1995):
SP (spare function), SC1 (schwefel’s problem 2.22), SC2
(schwefel’s problem 1.2), SC3 (schwefel’s problem 2.26),
GR (griewank function), AC (ackley function), RA (rastrign
function), and SH (shubert function). And “n” is the num-
ber of variables.
Function SUCC % EVAL # VAR
SP (n = 10) 100 5504.35 0.0
SC1 (n = 10)
100 4773.82 0.0
SC2 (n = 10)
44 5635.07 0.0
SC3 (n = 2)
92 2116.56 0.0
GR (n = 10)
100 5321.53 0.0
AC (n = 10)
100 5483.51 0.0
RA (n = 10)
92 7590.75 0.0
SH (n = 2)
38 4240.23 0.0013
where
ˆ
f refers to the best function value obtained by
GPSA, f
∗
refers to the known exact global minimum,
and ε
1
and ε
2
are small positive numbers. We set ε
1
and ε
2
equal to 10
−3
and 10
−6
, respectively. The re-
sults are shown in Table 1, where the average number
of function evaluations and the variance are related
only to successful trials. Table 1 shows that GPSA
reached the global minima in a very good success rate
for the majority of the tested functions. Moreover, the
numbers of function evaluations and the average er-
rors show the efﬁciency of the method.
On the other hands, GA had few successful tri-
als on any test functions at the termination point of
GPSA.
6 CONCLUSIONS
This paper ﬁrst developed a new class of pattern
search method that digitizes the patterns, called the
digital pattern search (DPS) method. Then, we pre-
sented a new hybrid global search algorithm, the ge-
netic pattern search algorithm (GPSA), which has a
self-adapting technique to modify the step size and
chase the approximate optimal direction. Applying
the DPS method in addition to the ordinary GA oper-
ators such as recombination and mutation enhances
the exploration process and accelerates the conver-
gence of the proposed algorithm. The experimental
results also showed that the GPSA works successfully
on some well known test functions.
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