Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) -4 computational complexity (where is the number of objectives and is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) solutions. Simulation results on five difficult test problems show that the proposed NSGA-II is able to find much better spread of solutions in all problems compared to PAES---another elitist multi-objective EA which pays special attention towards creating a diverse Pareto-optimal front. Because of NSGA-II's low computational requirements, elitist approach, and parameter-less sharing approach, NSGA-II should find increasing applications in the years to come.

The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Pareto-optimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) SPEA has shown very good performance in comparison to other multiobjective evolutionary algorithms, and therefore it has been a point of reference in various recent investigations, e.g., (Corne, Knowles, and Oates 2000). Furthermore, it has been used in different applications, e.g., (Lahanas, Milickovic, Baltas, and Zamboglou 2001). In this paper, an improved version, namely SPEA2, is proposed, which incorporates in contrast to its predecessor a fine-grained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method. The comparison of SPEA2 with SPEA and two other modern elitist methods, PESA and NSGA-II, on different test problems yields promising results. 1

In many problems of interest, performance can be evaluated using tests, such as examples in concept learning, test points in function approximation, and opponents in game-playing. Evaluation on all tests is often infeasible. Identification of an accurate evaluation or fitness function is a difficult problem in itself, and approximations are likely to introduce human biases into the search process. Coevolution evolves the set of tests used for evaluation, but has so far often led to inaccurate evaluation. We show that for any set of learners, a Complete Evaluation Set can be determined that provides ideal evaluation as specified by Evolutionary Multi-Objective Optimization. This provides a principled approach to evaluation in coevolution, and thereby brings automatic ideal evaluation within reach. The Complete Evaluation Set is of manageable size, and progress towards it can be accurately measured. Based on this observation, an algorithm named DELPHI is developed. The algorithm is tested on problems likely to permit progress on only a subset of the underlying objectives. Where all comparison methods result in overspecialization, the proposed method and a variant achieve sustained progress in all underlying objectives. These findings demonstrate that ideal evaluation may be approximated by practical algorithms, and that accurate evaluation for test-based problems is possible even when the underlying objectives of a problem are unknown.

In real world problems, one is often faced with the problem of multiple, possibly competing, goals, which should be optimized simultaneously. These competing goals give rise to a set of compromise solutions, generally denoted as Pareto-optimal. If none of the objectives have preference over the other, none of these trade-off solutions can be said to be better than any other solution in the set. Multi-objective Evolutionary Algorithms (MOEAs) can find these optimal trade-offs in order to get a set of solutions that are optimal in an overall sense.

Mu l ip often conflicting objectives arise naturalj in most real worl optimization scenarios. As evol tionaryalAxjO hms possess several characteristics that are desirabl e for this type of probl em, this clOv of search strategies has been used for mul tiobjective optimization for more than a decade. Meanwhil e evol utionary mul tiobjective optimization has become establ ished as a separate subdiscipl ine combining the fiel ds of evol utionary computation and cl assical mul tipl e criteria decision ma ing. This paper gives an overview of evol tionary mu l iobjective optimization with the focus on methods and theory. On the one hand, basic principl es of mu l iobjective optimization and evol tionary alA#xv hms are presented, and various al gorithmic concepts such as fitness assignment, diversity preservation, and el itism are discussed. On the other hand, the tutorial incl udes some recent theoretical resul ts on the performance of mu l iobjective evol tionaryalvDfifl hms and addresses the question of how to simpl ify the exchange of methods and appl ications by means of a standardized interface. 1

This study investigates the use of multiobjective techniques in Genetic Programming (GP) in order to evolve compact programs and to reduce the effects caused by bloating. The proposed approach considers the program size as a second, independent objective besides the program functionality. In combination with a recent multiobjective evolutionary technique, SPEA2, this method outperforms four other strategies to reduce bloat with regard to both convergence speed and size of the produced programs on a even-parity problem.

. This paper studies the influence of what are recognized as key issues

Over the past few years, the research on evolutionary algorithms

Clustering is a core problem in data-mining with innumerable applications spanning many fields. A key di#culty of e#ective clustering is that for unlabelled data a `good' solution is a somewhat ill-defined concept, and hence a plethora of valid measures of cluster quality have been devised. Most clustering algorithms optimize just one such objective (often implicitly) and are thus limited in their scope of application. In this paper, we investigate whether an EA optimizing a number of di#erent clustering quality measures simultaneously can find better solutions. Using problems where the correct classes are known, our results show a clear advantage to the multiobjective approach: it exhibits a far more robust level of performance than the classic k-means and average-link agglomerative clustering algorithms over a diverse suite of 15 real and synthetic data sets, sometimes outperforming them substantially.

In this paper we propose two novel approaches for solving constrained multi-objective optimization problems using steady state GAs.