Dominance Based Crossover Operator for Evolutionary Multi-objective Algorithms DRAFT VERSION

In spite of the recent quick growth of the Evolutionary Multi-objective Optimization (EMO) research field, there has been few trials to adapt the general variation operators to the particular context of the quest for the Pareto-optimal set. The only exceptions are some mating restrictions that take in account the distance between the potential mates – but contradictory conclusions have been reported. This paper introduces a particular mating restriction for Evolutionary Multi-objective Algorithms, based on the Pareto dominance relation: the partner of a non-dominated individual will be preferably chosen among the individuals of the population that it dominates. Coupled with the BLX crossover operator, two different ways of generating offspring are proposed. This recombination scheme is validated within the well-known NSGA-II framework on three bi-objective benchmark problems and one real-world bi-objective constrained optimization problem. An acceleration of the progress of the population toward the Pareto set is observed on all problems. Introduction The idea of restricted mating is not new in Evolutionary Computation: Goldberg [7] already suggested to forbid, or at least restrict, the crossover between too different individuals (i.e. that are too far apart for some distance on the genotypic space) – which makes sense for single-objective problems as soon as the population has started to accumulate on different fitness peaks, as recombining individuals from different peaks would often lead to lethal individuals. This idea has been transposed in the framework of Evolutionary Multi-objective Algorithms (EMAs) by Hajela and Lin [8], and by Fonseca and Fleming [6]. Nevertheless, Zitzler and Thiele [17] did not observe any improvement when mating similar individuals. On the other hand, Horn et al. [9] present an argument supporting mating of dissimilar individuals: in the multi-objective framework, because the population diversity is enforced, the information provided by very different solutions can be combined in such way that a new type of (good) compromises can hopefully be obtained. Nevertheless, Schaffer reported the absence of the improvement when mating dissimilar individuals. To sum up, no clear conclusion can be drawn from existing experiments on the usefulness of restricted mating based on the (dis)similarity between mates. On a real-world design problem, using a very specific representation, Wildman et Parks [13] have investigated different pairing strategies based on maximizing or minimizing different similarity measures. In particular, the similarity in the sense of the dominance rank has been considered, and enforcing the mating of the individuals from the elite archive with the individuals from the population, in an archive-based EMA, has been observed to be beneficial. However, in all studies enumerated above, the efficiency of the proposed mating restrictions has been measured by the quality of the final non-dominated solutions, without addressing the issue of computational time. In this paper, we propose a restricted mating strategy whose main effect is to accelerate the progress of the population of an EMA toward the Pareto set. The main idea is fairly simple, and consists in using the Pareto dominance relation when choosing a mate for the best (non-dominated) individuals. Note that a more detailed presentation (in French) can be found in [10]. The paper is organized as follows. Next section briefly introduces evolutionary multi-objective optimization, and describes in more detail the NSGAII algorithm, one of the best performing EMA to-date, that will be used in all experiments. Two slightly different implementations of the dominance-based crossover operator are then proposed in Section 2, based on BLX-α crossover, used throughout this study. Section 3 presents some experimental results witnessing the acceleration of the progress toward the Pareto set when using the proposed mating restrictions. Finally, Section 4 gives some guidelines for a more rigorous and complete validation of the proposed strategy, as well as for its possible refinements. 1 Evolutionary Multi-objective Optimization Multi-objective optimization aims at simultaneously optimizing several contradictory objectives. For such kind of problems, there does not exist a single optimal solution, and compromises have to be made. An element of the search space x is said to Pareto-dominate another element y if x is not worse than y with respect to all objectives, and is strictly better than y with respect to at least one objective. The set of all elements of the search space that are not Pareto-dominated by any other element is called the Pareto set of the multi-objective problem at hand: it represents the best possible compromises with respect to the contradictory objectives. Solving a multi-objective problem amounts to choose one solution among those non-dominated solutions, and some decision arguments have to be given. Unlike classical optimization methods, that generally find one of the Pareto optimal solutions by making the initial optimization problem single-objective, EMAs are to-date the only algorithms that directly search for the whole Pareto set, allowing decision makers to choose one of the Pareto solutions with more complete information. 1.1 Pareto-based Evolutionary Algorithms In order to find a good approximation of the Pareto set (a uniform and well spread sampling of the non-dominated solutions, close to the actual Pareto set of the problem at hand), EMAs have to enforce some progress toward the Pareto set while, at the same time, preserving diversity between the non-dominated solutions. Numerous evolutionary methods have been designed in the past years for the particular task of searching for the Pareto set (the interested reader will find a good summary in [2]). The best performing among them (NSGA-II [3], SPEA2 [16], PESA [1]) are directly based on the Pareto dominance relation, that actually ensures progressing toward the non-dominated set. Among the diversity preserving techniques, some were transposed to EMAs from single-objective EAs (such as sharing, for instance), while others, like the crowding distance described in next subsection, are specific to the multi-objective framework. Another recognized important feature of EMAs is elitism [15], directly related to the notion of the Pareto dominance in EMAs: the non-dominated individuals can be preserved either by maintaining an archive (SPEA2 and PESA) or by using a deterministic replacement procedure (NSGA-II).