This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having @math points to partition in @math clusters, the complexity is proven in @math (resp @math ) time and @math memory space for the continuous (resp discrete) @math -center problem. @math -center problems have complexities in @math . To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.
Selection or clustering points in PF have been studied with applications to MOO algorithms. Firstly, a motivation is to store representative elements of a large PF (exponential sizes of PF are possible @cite_45 ) for exact methods or population meta-heuristics. Maximizing the quality of discrete representations of Pareto sets was studied with the hypervolume measure in the Hypervolume Subset Selection (HSS) problem @cite_31 @cite_43 . Secondly, a crucial issue in the design of population meta-heuristics for MOO problems is to select relevant solutions for operators like cross-over or mutation phases in evolutionary algorithms @cite_2 @cite_38 . Selecting knee-points is another known approach for such goals @cite_42 .
@cite_38 @cite_42 @cite_43 @cite_45 @cite_2 @cite_31
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Recently, many meta-heuristic algorithms have been proposed to serve as the basis of a t-way test generation strategy (where t indicates the interaction strength) including Genetic Algorithms (GA), Ant Colony Optimization (ACO), Simulated Annealing (SA), Cuckoo Search (CS), Particle Swarm Optimization (PSO), and Harmony Search (HS). Although useful, meta-heuristic algorithms that make up these strategies often require specific domain knowledge in order to allow effective tuning before good quality solutions can be obtained. Hyper-heuristics provide an alternative methodology to meta-heuristics which permit adaptive selection and or generation of meta-heuristics automatically during the search process. This paper describes our experience with four hyper-heuristic selection and acceptance mechanisms namely Exponential Monte Carlo with counter (EMCQ), Choice Function (CF), Improvement Selection Rules (ISR), and newly developed Fuzzy Inference Selection (FIS), using the t-way test generation problem as a case study. Based on the experimental results, we offer insights on why each strategy differs in terms of its performance. Recently, the hybridization between evolutionary algorithms and other metaheuristics has shown very good performances in many kinds of multiobjective optimization problems (MOPs), and thus has attracted considerable attentions from both academic and industrial communities. In this paper, we propose a novel hybrid multiobjective evolutionary algorithm (HMOEA) for real-valued MOPs by incorporating the concepts of personal best and global best in particle swarm optimization and multiple crossover operators to update the population. One major feature of the HMOEA is that each solution in the population maintains a nondominated archive of personal best and the update of each solution is in fact the exploration of the region between a selected personal best and a selected global best from the external archive. Before the exploration, a selfadaptive selection mechanism is developed to determine an appropriate crossover operator from several candidates so as to improve the robustness of the HMOEA for different instances of MOPs. Besides the selection of global best from the external archive, the quality of the external archive is also considered in the HMOEA through a propagating mechanism. Computational study on the biobjective and three-objective benchmark problems shows that the HMOEA is competitive or superior to previous multiobjective algorithms in the literature. Given a nondominated point set of size and a suitable reference point , the Hypervolume Subset Selection Problem HSSP consists of finding a subset of size that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of . In contrast, the best upper bound available for is . Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of using a greedy strategy. In this article, greedy -time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem. Most problems encountered in practice involve the optimization of multiple criteria. Usually, some of them are conflicting such that no single solution is simultaneously optimal with respect to all criteria, but instead many incomparable compromise solutions exist. In recent years, evidence has accumulated showing that Evolutionary Algorithms (EAs) are effective means of finding good approximate solutions to such problems. One of the crucial parts of EAs consists of repeatedly selecting suitable solutions. In this process, the two key issues are as follows: first, a solution that is better than another solution in all objectives should be preferred over the latter. Second, the diversity of solutions should be supported, whereby often user preference dictates what constitutes a good diversity. The hypervolume offers one possibility to achieve the two aspects; for this reason, it has been gaining increasing importance in recent years. The present thesis investigates three central topics of the hypervolume that are still unsolved: 1: Although more and more EAs use the hypervolume as selection criterion, the resulting distribution of points favored by the hypervolume has scarcely been investigated so far. Many studies only speculate about this question, and in parts contradict one another. 2: The computational load of the hypervolume calculation sharply increases the more criteria are considered. This hindered so far the application of the hypervolume to problems with more than about five criteria. 3: Often a crucial aspect is to maximize the robustness of solutions, which is characterized by how far the properties of a solution can degenerate when implemented in practice. So far, no attempt has been made to consider robustness of solutions within hypervolume-based search.