Real-world optimization problems are often subject to uncertainties caused by, e.g., missing information in the problem domain or stochastic models. These uncertainties can take different forms in terms of distribution, bounds, and central tendency. In the multiobjective context, some approaches have been proposed to take uncertainties into account within the optimization process. Most of them are based on a stochastic extension of Pareto dominance that is combined with standard, non-stochastic diversity preservation mechanisms. Furthermore, it is often assumed that the shape of the underlying probability distribution is known and that for each solution there is a 'true' objective value per dimension which is disturbed by noise.In this paper, we consider a slightly different scenario where the optimization goal is specified in terms of a quality indicator-a real-valued function that induces a total preorder on the set of Pareto set approximations. We propose a general indicator-model that can handle any type of distribution representing the uncertainty, allows different distributions for different solutions, and does not assume a 'true' objective vector per solution, but in general regards a solution to be inherently associated with an unknown probability distribution in the objective space. To this end, several variants of an evolutionary algorithm for a specific quality indicator, namely the ǫ-indicator, are suggested and empirically investigated. The comparison to existing techniques such as averaging or probabilistic dominance ranking indicates that the proposed approach is especially useful for high-dimensional objective spaces. Moreover, we introduce a general methodology to visualize and analyze Pareto set approximations in the presence of uncertainty which extends the concept of attainment functions. I. MotivationKnowledge about the set of Pareto-optimal solutions is useful in many applications involving multiple objectives. Therefore, considerable research, particularly in the context of evolutionary computation, has been devoted to generating methods, i.e., techniques that try to generate the entire Pareto set or approximations of it. One recent approach is based on quality indicators where a function I assigns each Pareto set approximation a real value reflecting its quality, cf. (Zitzler et al. 2003); the goal is to identify a Pareto set approximation that minimizes (or maximizes) I. As such, I induces a total order of the set of approximation sets in the objective space, in contrast to the classical aggregation functions like weighted sum that operate on single solutions only and gives rise to a total order of the corresponding objective vectors. Since evolutionary multiobjective optimization using quality indicators is a relatively new concept, it is an open question how to deal with uncertainties in this framework. Many real-world optimization problems are subject to uncertainties and therefore this aspect needs to be accounted for. Among the different types of uncertainty one can distin...
Multi-objective optimization using evolutionary algorithms has been largely studied in the literature. Here, we propose formal methods to solve some problems appearing frequently in the design of such algorithms. To evaluate the effectiveness of the introduced mechanisms, we apply them to the flow-shop scheduling problem. We propose a dynamic mutation Pareto Genetic Algorithm (GA) in which different genetic operators are used simultaneously in an adaptive manner, taking into account the history of the search. We present a diversification mechanism which combines sharing in the objective space as well as in the decision space, in which the s u e of the niche is automatically calculated. We propose also an hybrid approach which combines the pareto GA with local search. Finally, we propose two performance indicators to evaluate t he effectiveness of the introduced mechanisms.
In recent years, the application of metaheuristic techniques to solve multi-objective optimization problems (MOPs) has become an active research area. Solving these kinds of problems involves obtaining a set of Pareto-optimal solutions in such a way that the corresponding Pareto front fulfills the requirements of convergence to the true Pareto front and uniform diversity. Most studies on metaheuristics for multiobjective optimization are focused on Evolutionary Algorithms, and some of the state-of-the-art techniques belong to this class of algorithms. Our goal in this paper is to study open research lines related to metaheuristics but focusing on less explored areas to provide new perspectives to those researchers interested in multi-objective optimization. In particular, we focus on non-evolutionary metaheuristics, hybrid multi-objective metaheuristics, parallel multi-objective optimization, and multi-objective optimization under uncertainty. We analyze these issues and discuss open research lines.
Abstract. This paper presents ParadisEO-MOEO, a white-box objectoriented generic framework dedicated to the flexible design of evolutionary multi-objective algorithms. This paradigm-free software embeds some features and techniques for Pareto-based resolution and aims to provide a set of classes allowing to ease and speed up the development of computationally efficient programs. It is based on a clear conceptual distinction between the solution methods and the multi-objective problems they are intended to solve. This separation confers a maximum design and code reuse. ParadisEO-MOEO provides a broad range of archive-related features (such as elitism or performance metrics) and the most common Pareto-based fitness assignment strategies (MOGA, NSGA, SPEA, IBEA and more). Furthermore, parallel and distributed models as well as hybridization mechanisms can be applied to an algorithm designed within ParadisEO-MOEO using the whole version of ParadisEO. In addition, GUIMOO, a platform-independant free software dedicated to results analysis for multi-objective problems, is briefly introduced.
Abstract. Path relinking algorithms have proved their efficiency in single objective optimization. Here we propose to adapt this concept to Pareto optimization. We combine this original approach to a genetic algorithm. By applying this hybrid approach to a bi-objective permutation flow-shop problem, we show the interest of this approach.In this paper, we present first an Adaptive Genetic Algorithm dedicated to obtain a first well diversified approximation of the Pareto set. Then, we present an original hybridization with Path Relinking algorithm, in order to intensify the search between solutions obtained by the first approach. Results obtained are promising and show that cooperation between these optimization methods could be efficient for Pareto optimization.
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