Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Paretooptimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modi�cations to the baseline algorithm are also suggested. The concept of-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

Abstract. This paper discusses how preference information of the decision maker can in general be integrated into multiobjective search. The main idea is to first define the optimization goal in terms of a binary performance measure (indicator) and then to directly use this measure in the selection process. To this end, we propose a general indicator-based evolutionary algorithm (IBEA) that can be combined with arbitrary indicators. In contrast to existing algorithms, IBEA can be adapted to the preferences of the user and moreover does not require any additional diversity preservation mechanism such as fitness sharing to be used. It is shown on several continuous and discrete benchmark problems that IBEA can substantially improve on the results generated by two popular algorithms, namely NSGA-II and SPEA2, with respect to different performance measures. 1

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multi-objective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. However, none of the multi-objective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper we discuss why a number of earlier MOEAs do not have such properties. A new archiving strategy is proposed that maintains a subset of the generated solutions. It guarantees convergence and diversity according to welldefined criteria, i.e. #-dominance and #-Pareto optimality.

The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a faster determination of the S-metric value is of essential importance. This paper describes how to consider the S-metric as a special case of a more general geometrical problem called Klee’s measure problem (KMP). For KMP, an algorithm exists with run time O(n logn + n d/2 log n), for n points of d ≥ 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications of the implementation are concerned that save on a factor of O(logn) and establish an upper bound of O(n logn + n d/2) for the S-metric calculation, improving the previously known bound of O(n d−1).

Many modern multiobjective evolutionary algorithms (MOEAs) store the points discovered during optimization in an external archive, separate from the main population, as a source of innovation and/or for presentation at the end of a run. Maintaining a bound on the size of the archive may be desirable or necessary for several reasons, but choosing which points to discard and which to keep in the archive, as they are discovered, is not trivial. In this paper we briefly review the state-of-the-art in bounded archiving, and present a new method based on locally maximizing the hypervolume dominated by the archive. The new archiver is shown to outperform existing methods, on several problem instances, with respect to the quality of the archive obtained when judged using three distinct quality measures.

The classic NFL theorems are invariably cast in terms of single objective optimization problems. We confirm that the classic NFL theorem holds for general multiobjective fitness spaces, and show how this follows from a 'single-objective' NFL theorem. We also show that, given any particular Pareto Front, an NFL theorem holds for the set of all multiobjective problems which have that Pareto Front. It follows that, given any 'shape' or class of Pareto fronts, an NFL theorem holds for the set of all multiobjective problems in that class. These findings have salience in test function design. Such NFL results are cast in the typical context of absolute performance, assuming a performance metric which returns a value based on the result produced by a single algorithm. But, in multiobjective search...

In this paper we are concerned with finding the Pareto optimal front or a good approximation to it. Since non-dominated solutions represent the goal in multiobjective optimisation, the dominance relation is frequently used to establish preference between solutions during the search. Recently, relaxed forms of the dominance relation have been proposed in the literature for improving the performance of multiobjective search methods. This paper investigates the influence of different fitness evaluation methods on the performance of two multiobjective methodologies when applied to a highly constrained two-objective optimisation problem. The two algorithms are: the Pareto archive evolutionary strategy and a population-based annealing algorithm. We demonstrate here, on a highly constrained problem, that the method used to evaluate the fitness of candidate solutions during the search affects the performance of both algorithms and it appears that the dominance relation is not always the best method to use.

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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 – This study introduces a hybrid multiobjective evolutionary algorithm (MOEA) for the optimization of aircraft control system design. The strategy suggested here is composed mainly of two stages. The first stage consists of training an Artificial Neural Network (ANN) with objective values as inputs and decision variables as outputs to model an approximation of the inverse of the objective function used. The second stage consists of a local improvement phase in objective space preserving objectives relationships, and a mapping process to decision variables using the trained ANN. Both the hybrid MOEA and the original MOEA were applied to an aircraft control system design application for assessment. 1