Differential Evolution Competitive Setting of Control Parameters

Abstract. This paper is focused on the adaptation of control parameters in differential evolution. The competition of different control parameter settings was proposed in order to ensure the selfadaptation of parameters within the search process. Several variants of such algorithm were tested on six functions at four levels of search-space dimension. This competitive differential evolution proved to be more reliable and less time-consuming than the standard differential evolution. The competitive differential evolution also outperformed other algorithms that were tested.
1
IntroLeabharlann uctionD ⊂ Rd ,
We will deal with the global optimization problem: for a given objective function f : D → R,
the point x∗ is to be found such that x∗ = arg minx∈D f (x). The point x∗ is called the global minimum point and D is the search space. We focus on the problems, where the objective function is continuous and the search space is closed compact set, D = d i=1 [ai , bi ], ai < bi , i = 1, 2, . . . , d (box constrains). The problem of the global optimization is hard and plenty of stochastic algorithms were proposed for its solution, see e.g. [2, 11]. The authors of many such stochastic algorithms claim the efficiency and the reliability of searching for the global minimum. The reliability means that the point with minimal function value found in the search process is sufficiently close to the global minimum point and the efficiency means that the algorithm finds a point sufficiently close to the global minimum point at reasonable time. However, when we use such algorithms, we face the problem of the setting their control parameters. The efficiency and the reliability of many algorithms is strongly dependent on the values of control parameters. Recommendations given by authors are often vague or uncertain, see e.g. [12, 22]. A user is supposed to be able to change the parameter values according to the results of trial-and-error preliminary experiments with the search process. Such attempt is not acceptable in tasks, where the global optimization is one step on the way to the solution of the user’s problem or when the user has no experience in fine art of control parameter tuning. Adaptive robust algorithms reliable enough at reasonable time-consumption without the necessity of fine tuning their input parameters have been studied in recent years. The proposal of an adaptive generator of robust algorithms is described in Deb [4]. Winter et al. [17] proposed a flexible evolutionary agent for real-coded genetic algorithms. Theoretical analysis done by Wolpert and Macready implies, that any search algorithm cannot outperform the others for all objective functions [18]. In spite of this fact, there is empirical evidence, that some algorithms can outperform others for relatively wide class of problems both in the convergence rate and in the reliability of finding the global minimum point. Thus, the way to the adaptive algorithms leads rather trough the experimental research than trough purely theoretical approach.
There are several variants how to generate the mutant point u. One of the most popular (called DER in this text) generates the point u by adding the weighted difference of two points u = r1 + F (r 2 − r 3 ) , (1)
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Differential Evolution and its Control Parameters
The differential evolution (DE) works with two population P and Q of the same size N . The algorithm in pseudo-code is written as Algorithm 1. A new trial point y is composed of the current point xi of old population and the point u obtained by using mutation. If f (y ) < f (xi ) the point y is inserted into the new population Q instead of xi . After completion of the new population Q the old population P is replaced by Q and the search continues until stopping condition is fulfilled.
Proceedings of the International Multiconference on Computer Science and Information Technology pp. 207–213
ISSN 1896-7094 c 2006 PIPS
Differential Evolution: Competitive Setting of Control Parameters⋆
where r 1 , r 2 and r3 are three distinct points taken randomly from P (not coinciding with the current xi ) and F > 0 is an input parameter. Another variant called DEBEST generates the point u according to formula u = xmin + F (r 1 + r2 − r3 − r 4 ) , (2) where r 1 , r2 , r 3 , r 4 are four distinct points taken randomly from P (not coinciding with the current xi ), xmin is the point of P with minimal function value, and F > 0 is an input parameter. The elements yj , j = 1, 2. . . . , d of trial point y are built up by the crossover of its parents xi and u using the following rule uj if Uj ≤ C or j=l yj = (3) xij if Uj > C and j = l , where l is a randomly chosen integer from {1, 2, . . . , d}, U1 , U2 , . . . , Ud are independent random variables uniformly distributed in [0, 1), and C ∈ [0, 1] is an input parameter influencing the number of elements to be exchanged by crossover. Eq. (3) ensures that at least one element of xi is changed even if C = 0. The differential evolution has become one of the most popular algorithms for the continuous global optimization problems in recent years, see [5]. But it is known that the efficiency of the search for the global minimum is very sensitive to the setting of values F and C . The recommended values are F = 0.8 and C = 0.5, but even Storn and Price in their principal paper [12] use 0.5 ≤ F ≤ 1 and 0 ≤ C ≤ 1 depending on the results of preliminary tuning. They also set the size of population less than the recommended N = 10 d in many of their test tasks. Many papers deal with the setting of control parameters for differential evolution. Ali and T¨ orn [1] suggested to adapt the value of the scaling factor F within the search process according to the equation F =