Fuzzy c-means clustering-based mating restriction for multiobjective optimization
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ORIGINAL ARTICLE
Fuzzy c-means clustering-based mating restriction for multiobjective optimization Yi Zhang1 · Zimu Li1 · Hu Zhang2 · Zhen Yu1 · Tongtong Lu3
Received: 19 January 2016 / Accepted: 28 March 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Mating restriction is an important approach to improve the performance of multiobjective evolutionary algorithms (MOEAs). This paper designs a fuzzy c-means clustering-based mating restriction (FMR) scheme, and proposes a fuzzy c-means clustering-based MOEA named as FCMMO. FMR employs a fuzzy c-means clustering algorithm to discover the clustering structure of the solutions in the population. Based on the structure, a specific mating pool is determined for each solution to generate new solutions. Experimental studies show that FCMMO outperforms five state of the art MOEAs on a set of test instances with complicated Pareto front shapes and Pareto set structures, and FMR significantly contributes to the good performance of FCMMO. Keywords Evolutionary algorithm · Multiobjective optimization · Mating restriction · Fuzzy C-means clustering
1 Introduction In engineering, there exist a variety of optimization problems with multiple objectives. This type of optimization problems are called multi-objective optimization problems * Hu Zhang [email protected] 1
College of Mechanical and Power Engineering, China Three Gorges University, Yi Chang 443002, China
2
Beijing Electro-mechanical Engineering Institute, Beijing 100074, China
3
College of Economy and Management, China Three Gorges University, Yi Chang 443002, China
(MOPs). A box-constrained MOP is formulated as follows [25].
[ ]T min 𝐅(𝐱) = f1 (𝐱), f2 (𝐱), … , fm (𝐱) (1) ( )T s.t. 𝐱 = x1 , x2 , … , xn ∈ Ω ( )T where 𝐱 = x1, x2�, ⋯ , x�n is a vector of n decision vari∏n ables; Ω = i=1 ai , bi ⊆ Rn defines the decision (search) space; ai and bi are the lower and upper boundaries of variable xi, respectively; 𝐅:Ω → Rm represents the mapping from search space to objective space, where m objective functions fi (𝐱), i = 1, … , m are to be considered. In general, the optimization objectives of a MOP are conflicted, thus there does not exist a single solution that can optimize all the objectives simultaneously. Consequently, a set of trade-off solutions that can compromise ( all the objec)T 𝐮 = u1 , u2 , … , um , tives( exist for a MOP. Suppose that )T 𝐯 = v1 , v2 , … , vm are two vectors. If ui ⩽ vi for all i ∈ {1, 2, … , m}, but there exists at least one index j, such that uj < vj, then 𝐮 is said to dominate 𝐯, denoted by 𝐮 ≺ 𝐯. Based on the dominance, in a MOP, a solution 𝐱∗ ∈ Ω is called Pareto optimal solution if there is no 𝐱 ∈ Ω such that 𝐅(𝐱) ≺ 𝐅(𝐱∗ ). The set of all the Pareto optimal solutions are named as Pareto set (PS). The set of the objective values of all the Pareto optimal solutions constitute Pareto front (PF). Since it is impossible to find all the Pareto optimal solutions of a MOP, therefore, the solver of MOP is usually aimed to obtain a set of approximated solutions whose objective val
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