Minimizing the Real Functions of the ICEC'96 Contest by Differential Evolution

by adding the weighted difference between two population vectors to a third vector. If the resulting vector yields a lower objective function value than a predetermined population member, the newly generated vector will replace the vector with which it was compared in the following generation; otherwise, the old vector is retained. This basic principle, however, is extended when it comes to the practical variants of DE. For example an existing vector can be perturbed by adding more than one weighted difference vector to it. In most cases, it is also worthwhile to mix the parameters of the old vector with those of the perturbed one. The performance of the resulting vector is then compared to that of the old vector. We will describe two variants of DE which have proven to be useful. Differential Evolution (DE) has recently proven to be an efficient method for optimizing real-valued multi-modal objective functions. Besides its good convergence properties and suitability for parallelization, DE's main assets are its conceptual simplicity and ease of use. This paper describes two variants of DE and summarizes their performance on the real test functions of the ICEC'96 contest. Introduction Differential Evolution (DE) [1], [2] has proven to be a promising candidate for optimizing realvalued multi-modal objective functions. Besides its good convergence properties DE is very simple to understand and to implement. DE is also particularly easy to work with, having only a few control variables which remain fixed throughout the entire optimization procedure. Variant DE/rand/1 For each vector x i G , , i = 0,1,2,...,NP-1, a perturbed vector v i G , +1 is generated according DE is a parallel direct search method which utilizes NP D-dimensional parameter vectors: to v x F x x i G r G r G r G , , , , ( ) + = + ⋅ − 1 3 1 2 (2) xi,G, i = 0, 1, 2, ... , NP-1, (1) with r r r NP 1 2 3 0 1 , , , ∈ − , integer and mutually different, and F > 0. as a population for each generation G, i.e. for each iteration of the optimization. NP doesn't change during the minimization process. The initial population is chosen randomly and should try to cover the entire parameter space uniformly. As a rule, we will assume a uniform probability distribution for all random decisions unless otherwise stated. The crucial idea behind DE is a scheme for generating trial parameter vectors. Basically, DE generates new parameter vectors The integers r1, r2 and r3 are chosen randomly from the interval [1, NP] and are different from the running index i. F is a real and constant factor ∈ [0, 2] which controls the amplification of the differential variation ( ) , , x x r G r G 2 3 − . Note that the vector x r G 1, which is perturbed to yield v i G , +1has no relation to x i G , but is a randomly chosen population member. Fig. 1 is a two dimensional example that illustrates the different vectors which play a part in the vector-generation scheme. The notation: DE/rand/1 specifies that the vector to be perturbed is randomly chosen and that the perturbation consists of one weighted difference vector. rand() is supposed to generate a random number ∈ [0,1):