Solving Multi-Objective Optimization Problems using Differential Evolution and a Maximin Selection Criterion Adriana Menchaca-Mendez CINVESTAV-IPN, Departamento de Computaci´ on Av. IPN 2508. San Pedro Zacatenco M´ exico D.F. 07300, M ´ EXICO, email: adriana.menchacamendez@gmail.com Carlos A. Coello Coello CINVESTAV-IPN, Departamento de Computaci´ on Av. IPN 2508. San Pedro Zacatenco M´ exico D.F. 07300, M ´ EXICO, email: ccoello@cs.cinvestav.mx Abstract—In this paper, we propose a new selection operator (based on a maximin scheme and a clustering technique), which is incorporated into a differential evolution algorithm to solve multi-objective optimization problems. The resulting algorithm is called Maximin-Clustering Differential Evolution (MCDE) and, is validated using standard test problems and performance measures taken from the specialized literature. Our preliminary results indicate that MCDE is able to outperform NSGA-II and that is competitive with a hypervolume-based approach (SMS- EMOA), but at a significantly lower computational cost. I. I NTRODUCTION Many optimization problems arising in the real world involve multiple objective functions which must be satisfied simultaneously. They are generically called multiobjective optimization problems (MOPs) and usually their objectives are in conflict. In MOPs, the notion of optimality refers to the best possible trade-offs among the objectives. Consequently, no single solution exists, but several (the so-called Pareto optimal set whose image is called the Pareto front). When ap- plying evolutionary algorithms to solve MOPs, we normally have two main goals [1]: (i) to find solutions that are, as close as possible, to the true Pareto front and, (ii) to produce solutions that are spread along the Pareto front as uniformly as possible. When studying multi-objective evolutionary algorithms (MOEAs), we find two main types of approaches: (i) those that incorporate the concept of Pareto optimality in their selection mechanism, and (ii) those that do not use Pareto dominance to select individuals. Although the use of Pareto-based selection (mainly through the use of some Pareto ranking scheme [1]) has been the most popular choice within the specialized literature for the last 15 years, such type of approach has several limitations. From them, its poor scalability (when increasing the number of objectives) is, perhaps, the most remarkable. The quick increase in the number of nondominated solutions as we increase the number of objectives, rapidly dilutes the effect of the selection mechanism of a MOEA [2]. This has triggered an important amount of research on the so-called “many-objective optimization”, which refers to the study of problems having four or more objective functions. In the current literature, we can identify three main ap- proaches to cope with many-objectives problems, namely: (i) to adopt or propose a preference relation that induces a finer grain order on the solutions than that induced by the Pareto dominance relation [3], [4], [5], [6], (ii) to reduce the number of objectives of the problem during the search process [7] or, a posteriori, during the decision making process [8], [9], and, (iii) to adopt a selection scheme that does not rely on Pareto optimality (e.g., using compromise functions [10], alternative ranking schemes [11] or a selection mechanism based on a performance measure (from which hypervolume 1 has been a popular choice, in spite of its considerably high computational cost [13], [14]). Here, we study an approach from the third class, using differential evolution as our search engine. The main motivation of this work is to propose an alternative selection mechanism for MOEAs that can properly deal with many-objective optimization problems at a reasonably low computational cost. The focus of our study is the maximin fitness function [15]. This technique assigns a fitness to each individual in the population without using the concept of Pareto optimality. This scheme encompasses a guidance mechanism based on very simple (and computationally efficient) operations. Our preliminary study of this approach has indicated its suitability as a selection operator in a MOEA whose search engine adopts differential evolution [16], even in the presence of a high number of objectives. However, its lack of an appropri- ate diversity maintenance mechanism makes it inappropriate with respect to state-of-the-art MOEAs, which led us to propose the incorporation of a clustering technique. The proposed approach, called Maximin-Clustering Differential Evolution (MCDE) is validated with several standard test problems and performance measures. As will be seen later on, our proposed MCDE is able to outperform NSGA-II [17] and is competitive with a state-of-the-art hypervolume-based MOEA (SMS-EMOA) [14], but requiring a much lower computational cost. The remainder of this paper is organized as follows. Section II states the problem of our interest. The maximin fitness function is briefly described in Section III. Section IV describes in detail the selection operator that we propose and 1 The hypervolume (also known as the S metric or the Lebesgue Measure) of a set of solutions measures the size of the portion of objective space that is dominated by those solutions collectively [12]. U.S. Government work not protected by U.S. copyright WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IEEE CEC