Discrete Set Handling - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 173

expressions and AP uses it to synthesize the resulting function-program for cost func-
tion evaluation.
AP was evaluated in three versions. These three versions utilize for program syn-
thesis the same set of functions, terminals, etc., as in GP [9, 10]. The second version
labelled as AP meta (the first version, AP basic ) is modified in the sense of constant estima-
tion. For example, the so-called sextic problem was used in [9] to randomly generate
constants, whereas AP here uses only one, called K, which is inserted into Equation
7.11 below at various places by the evolutionary process. When a program is synthe-
sized, all Ks are indexed as K 1 , K 2 ,. , K n to obtain Equation 7.12 in the formula, and
then all K n are estimated by using a second evolutionary algorithm, the result of which
can be, for example, Equation 7.13. Because EA (slave) works under EA (master), i.e.,
EA master → program → K indexing → EA slave → estimation of K n , this version is called

AP with metaevolution, denoted as AP meta .

x 2 + K
π

(7.11)

K

x 2 + K 1
π

(7.12)

K 2

x 2 + 3 . 56
π − 229

(7.13)

Because this version is quite time-consuming, AP meta was further modified to the
third version, which differs from the second one in the estimation of K. This is accom-
plished by using a suitable method for nonlinear fitting (denoted AP nf ). This method has
shown the most promising performance when unknown constants are present. Results
of some comparative simulations can be found in [25, 26, 27].

7.4 Differential Evolution in Mathematica Code

Differential Evolution used in all experiments reported in this chapter has been based
on the Mathematica Programming environment. The aim of this part is to describe the
structure of the DE code and final code development. Source codes reported here are
only a part of fully developed notebook in environment Mathematica. Only the main
ideas and some parts of the final code are described here.
For the beginning of DE code development, it is important to prepare the popula-
tion and set all DE algorithm parameters like F, CR, NP and Generation. Population is
initialized as shown in Fig 7.11.

In[25]:= Population DoPopulation NP, Specimen

534.695, 442.422, 188.47 , 191.21, 194.845, 437.931 ,
70.135, 127.976, 99.3825 , 208.07, 214.324, 244.138 ,
41.6243, 236.027, 128.204 , 161.461, 335.943, 355.91 ,
106.047, 317.752, 402.607 , 157.266, 119.503, 163.852 ,
464.407, 507.525, 502.251 , 62.8734, 160.401, 149.99

Out[25]=

Fig. 7.11. Population initialization