Abstract not found

Indicator-based algorithms have become a very popular approach to solve multi-objective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this common approach with respect to the achieved multiplicative approximation ratio for a given multi-objective problem and relate it to a set of µ points on the Pareto front that achieves the best possible approximation ratio. For the class of linear fronts and a class of concave fronts, we prove that the hypervolume gives the best possible approximation ratio. In addition, we examine Pareto fronts of different shapes by numerical calculations and show that the approximation computed by the hypervolume may differ from the optimal approximation ratio.

Abstract. Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in higher dimensional objective spaces, a comprehensive benchmarking is presented, featuring several state-ofthe-art EMOA, as well as an aggregative approach and a restart strategy on established scalable test problems with three to six objectives. It is demonstrated why the performance of well-established EMOA (NSGA-II, SPEA2) rapidly degradates with increasing dimension. Newer EMOA like ε-MOEA, MSOPS, IBEA and SMS-EMOA cope very well with highdimensional objective spaces. Their specific advantages and drawbacks are illustrated, thus giving valuable hints for practitioners which EMOA to choose depending on the optimization scenario. Additionally, a new method for the generation of weight vectors usable in aggregation methods is presented. 1

Abstract Indicator-based methods to tackle multiobjective problems have become popular recently, mainly because they allow to incorporate user preferences into the search explicitely. Multiobjective Evolutionary Algorithms (MOEAs) using the hypervolume indicator in particular showed better performance than classical MOEAs in experimental comparisons. In this paper, the use of indicatorbased MOEAs is investigated for the first time from a theoretical point of view. We carry out running time analyses for an evolutionary algorithm with a (µ + 1)selection scheme based on the hypervolume indicator as it is used in most of the recently proposed MOEAs. Our analyses point out two important aspects of the search process. First, we examine how such algorithms can approach the Pareto front. Later on, we point out how they can achieve a good approximation for an exponentially large Pareto front. 1

contributions

The multiobjective evolutionary algorithm SMS-EMOA was shown by Emmerich et al. to outperform the well-established NSGA-II on a range of common test problems, using the S metric as comparison criterion. This study assesses which of the two algorithms performs best with respect to the optimization of water distribution networks, using the unconstrained three objective reformulation by Formiga et al. of this originally constrained single objective problem. The well-known test problems Two Loop and Hanoi have been examined, showing the best results by SMS-EMOA in both cases. 1

An open problem in multiobjective optimization using the Pareto optimality criteria, is how to evaluate the performance of different evolutionary algorithms that solve multi– objective problems. As the output of these algorithms is a non–dominated set (NS), this problem can be reduced to evaluate what NS is better than the others based on their projection on the objective space. In this work we propose a new performance measure for the evaluation of non–dominated sets, that ranks a set of NSs based on their convergence and dispersion. Its evaluations of the NSs agree with intuition. Also, we introduce a benchmark of test cases to evaluate performance measures, that considers several topologies of the Pareto Front.

Approximating the least hypervolume contributor: NP-hard in general,

The multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) is an evolutionary algorithm for continuous vector-valued optimization. It combines indicator-based selection based on the contributing hypervolume with the efficient strategy parameter adaptation of the elitist covariance matrix adaptation evolution strategy (CMA-ES). Step sizes (i.e., mutation strengths) are adapted on individual-level using an improved implementation of the 1/5-th success rule. In the original MO-CMA-ES, a mutation is regarded as successful if the offspring ranks better than its parent in the elitist, rank-based selection procedure. In contrast, we propose to regard a mutation as successful if the offspring is selected into the next parental population. This criterion is easier to implement and reduces the computational complexity of the MO-CMA-ES, in particular of its steady-state variant. The new step size adaptation improves the performance of the MO-CMA-ES as shown empirically using a large set of benchmark functions. The new update scheme in general leads to larger step sizes and thereby counteracts premature convergence. The experiments comprise the first evaluation of the MO-CMA-ES for problems with more than two objectives.

Abstract not found