An Adaptive Constraint Handling Technique for Dierential Evolution in Engineering Optimization

1. AbstractDiﬀerential Evolution is a simple and eﬃcient stochastic, population-based heuristics for global opti-mization over continuous spaces. As with other nature inspired techniques, there is no provision forconstraint handling in its original formulation, and a few possibilities have been proposed in the litera-ture. In this paper an adaptive penalty technique (APM), previously developed and applied to geneticalgorithms, is considered for constraint handling within diﬀerential evolution. The technique requires noextra parameters. Based on feedback obtained from the current status of the population of candidatesolutions, the technique automatically deﬁnes, for each constraint, its corresponding penalty coeﬃcient. Equality as well as inequality constraints can be dealt with. The APM technique, which has been shownto be quite eﬀective within genetic algorithms, is adapted here to the diﬀerential evolution context. Inorder to assess the applicability and performance of the proposed constraint handling scheme, severaltest-problems from the structural and mechanical engineering optimization literature are considered.2. Keywords: Diﬀerential Evolution, Constrained Optimization, Adaptive Penalty.3. IntroductionDiﬀerential Evolution (DE) is an evolutionary algorithm, proposed by Storn and Price [1], aimed atsolving continuous optimization problems. It is a simple, stochastic, population-based technique thatwas initially formulated to deal with unconstrained problems.One of the main features of DE is its low number of parameters to be set by the user: the populationsize (POP), the maximum number of generations (GEN), the recombination rate (CR), and the muta-tion rate (F). The population, often initially randomly generated, is evolved by the application of linearcombinations of individuals in the population. The basic operation performed is the addition to eachdesign variable in a given candidate solution of a term which is the diﬀerence between the values of suchvariable in other candidate solutions in the population. The number of individuals that contribute tothe diﬀerential, the way of the individuals are selected and the distribution of recombination determinethe model of DE. In this paper we use a model described in [2] that modiﬁed the DE/rand/1/bin (theindividuals for mutation are selected at random, just one pair of solutions is used to the calculate thescaled diﬀerence and a binomial recombination is chosen) in order to incorporate information of the bestsolution in the current population (but not as the DE/best/1/bin does) and also information of thecurrent parent to deﬁne the new search directions by using the DE mutation operator.DE is a simple and fast algorithm for continuous unconstrained optimization problems and, in order toextend its domain of application to practical problems in engineering, several modiﬁcations were pro-posed, specially to handle constrained problems with discrete, continuous, and mixed discrete-continuousvariables, as described in the section 4. One of them was the use of penalty methods to handle infeasibleindividuals, as proposed in [3]. Due to the good results obtained by the adaptive penalty method (APM)[4, 5] when applied within a genetic algorithm, we propose in this paper the use of this penalty techniquetogether with the DE. Furthermore, as the APM does not require any additional parameter, the resultingDE+APM technique does not require any parameter in addition to the standard DE parameters.4. A Brief Overview on DEThe original proposal of DE by Storm and Price [1] presents a simple and eﬃcient algorithm forglobal optimization over continuous spaces. The algorithm (1), from [2], shows the pseudo-code forthe DE/rand/1/bin variant. In this model the individuals involved in the mutation process are selectedrandomly and generate a single oﬀspring after the process of mutation and recombination. The DE1