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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In many multiobjective optimization problems, the Pareto Fronts and Sets contain a large number of solutions and this makes it difficult for the decision maker to identify the preferred ones. A possible way to alleviate this difficulty is to present to the decision maker a subset of a small number of solutions representatives of the Pareto Front characteristics. In this paper, a two-steps procedure is presented, aimed at identifying a limited number of representative solutions to be presented to the decision maker. Pareto Front solutions are first clustered into "families", which are then synthetically represented by a "head-of-the-family" solution. Level Diagrams are then used to represent, analyse and interpret the Pareto Front reduced to its head-of-the-family solutions. The procedure is applied to a reliability allocation case study of literature, in decision-making contexts both without or with explicit preferences by the decision maker on the objectives to be optimized. The current boom of Unmanned Aerial Vehicles (UAVs) is increasing the number of potential industrial and research applications. One of the most demanded topics in this area is related to the automated planning of a UAVs swarm, controlled by one or several Ground Control Stations (GCSs). In this context, there are several variables that influence the selection of the most appropriate plan, such as the makespan, the cost or the risk of the mission. This problem can be seen as a Multi-Objective Optimization Problem (MOP). On previous approaches, the problem was modelled as a Constraint Satisfaction Problem (CSP) and solved using a Multi-Objective Genetic Algorithm (MOGA), so a Pareto Optimal Frontier (POF) was obtained. The main problem with this approach is based on the large number of obtained solutions, which hinders the selection of the best solution. This paper presents a new algorithm that has been designed to obtain the most significant solutions in the POF. This approach is based on Knee Points applied to MOGA. The new algorithm has been proved in a real scenario with different number of optimization variables, the experimental results show a significant improvement of the algorithm performance. One way of solving multiple objective mathematical programming problems is finding discrete representations of the efficient set. A modified goal of finding good discrete representations of the efficient set would contribute to the practicality of vector maximization algorithms. We define coverage, uniformity and cardinality as the three attributes of quality of discrete representations and introduce a framework that includes these attributes in which discrete representations can be evaluated, compared to each other, and judged satisfactory or unsatisfactory by a Decision Maker. We provide simple mathematical programming formulations that can be used to compute the coverage error of a given discrete representation. Our formulations are practically implementable when the problem under study is a multiobjective linear programming problem. We believe that the interactive algorithms along with the vector maximization methods can make use of our framework and its tools. A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics. Optimizing the hypervolume indicator within evolutionary multiobjective optimizers has become popular in the last years. Recently, the indicator has been generalized to the weighted case to incorporate various user preferences into hypervolume-based search algorithms. There are two main open questions in this context: (i) how does the specified weight influence the distribution of a fixed number of points that maximize the weighted hypervolume indicator? (ii) how can the user articulate her preferences easily without specifying a certain weight distribution function? In this paper, we tackle both questions. First, we theoretically investigate optimal distributions of μ points that maximize the weighted hypervolume indicator. Second, based on the obtained theoretical results, we propose a new approach to articulate user preferences within biobjective hypervolume-based optimization in terms of specifying a desired density of points on a predefined (imaginary) Pareto front. Within this approach, a new exact algorithm based on dynamic programming is proposed which selects the set of μ points that maximizes the (weighted) hypervolume indicator. Experiments on various test functions show the usefulness of this new preference articulation approach and the agreement between theory and practice.