Combining Manhattan and Crowding Distances in Decision Space for Multimodal Multi-objective Optimization Problems
This paper presents a new variant of the Non-dominated Sorting Genetic Algorithm to solve Multimodal Multi-objective optimization problems. We introduce a novel method to augment the diversity of solutions in decision space by combining the Manhattan and
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Combining Manhattan and Crowding Distances in Decision Space for Multimodal Multi-objective Optimization Problems Mahrokh Javadi, Cristian Ramirez-Atencia, and Sanaz Mostaghim Abstract This paper presents a new variant of the Non-dominated Sorting Genetic Algorithm to solve Multimodal Multi-objective optimization problems. We introduce a novel method to augment the diversity of solutions in decision space by combining the Manhattan and crowding distance. In our experiments, we use six test problems with different levels of complexity to examine the performance of our proposed algorithm. The results are compared with NSGA-II and NSGA-II-WSCD algorithms. Using IGDX and IGD performance indicators, we demonstrate the superiority of our proposed method over the rest of competitors to provide a better approximation of the Pareto Set (PS) while not getting much worse results in objective space. Keywords Multimodality · Multi-modal problems · Multi-objective optimization · Evolutionary algorithms · Solution space diversity
9.1 Introduction In real world, there are many Multi-objective Optimization Problems (MOP) with at least two conflicting objectives in nature. This means that improving one of the objectives leads to deteriorating the value for the other objectives. Without loss of generality, a multi-objective minimization problems is formulated as follows:
M. Javadi (B) · C. Ramirez-Atencia · S. Mostaghim Faculty of Computer Science, Otto von Guericke University, Magdeburg, Germany e-mail: [email protected] C. Ramirez-Atencia e-mail: [email protected] S. Mostaghim e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Gaspar-Cunha et al. (eds.), Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences, Computational Methods in Applied Sciences 55, https://doi.org/10.1007/978-3-030-57422-2_9
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minimize f(x) = (f1 (x), f2 (x), . . . , fM (x)) subject to
(9.1)
x∈S⊂R gi (x) ≤ 0, i = 1, 2, . . . , G hj (x) = 0, j = 1, 2, . . . , H D
where x = (x1 , x2 , . . . , xD ) is considered as a D–dimensional decision vector and (f1 , f2 , . . . , fM ) is a M –dimensional objective vector. gi (x) and hj (x) are inequality and equality constraint functions in decision space. In order to deal with these problems, the concept of domination can be used. Given two vectors x, y ∈ S, x is said to be dominated by y (denoted by y ≺ x) if and only if ∀j ∈ {1, . . . , M }, fj (y) ≤ fj (x), and ∃k ∈ {1, . . . , M }, fk (y) < fk (x) [1]. The solution of multi-objective optimization problems, is a set of non-dominated solutions called Pareto-Set (PS), which the corresponding set of these solutions in the objective space is called the Pareto-Front (PF). In Multimodal Multi-Objective Optimization Problems, there are two or more distinct solutions in the PS, which correspond to the same value in the PF. In this area, most of the available literature deals
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