564 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 22, NO. 4, AUGUST 2018 A Constrained Decomposition Approach With Grids for Evolutionary Multiobjective Optimization Xinye Cai, Member, IEEE, Zhiwei Mei, Zhun Fan, Senior Member, IEEE, and Qingfu Zhang, Fellow, IEEE Abstract—Decomposition-based multiobjective evolutionary algorithms (MOEAs) decompose a multiobjective optimization problem (MOP) into a set of scalar objective subproblems and solve them in a collaborative way. Commonly used decompo- sition approaches originate from mathematical programming and the direct use of them may not suit MOEAs due to their population-based property. For instance, these decomposition approaches used in MOEAs may cause the loss of diversity and/or be very sensitive to the shapes of Pareto fronts (PFs). This paper proposes a constrained decomposition with grids (CDG) that can better address these two issues thus more suitable for MOEAs. In addition, different subproblems in CDG defined by the constrained decomposition constitute a grid system. The grids have an inherent property of reflecting the information of neighborhood structures among the solutions, which is a desirable property for restricted mating selection in MOEAs. Based on CDG, a constrained decomposition MOEA with grid (CDG-MOEA) is further proposed. Extensive experiments are conducted to compare CDG-MOEA with the domination- based, indicator-based, and state-of-the-art decomposition-based MOEAs. The experimental results show that CDG-MOEA out- performs the compared algorithms in terms of both the con- vergence and diversity. More importantly, it is robust to the shapes of PFs and can still be very effective on MOPs with complex PFs (e.g., extremely convex, or with disparately scaled objectives). Index Terms—Constrained decomposition, evolutionary multiobjective optimization, grids, robust to Pareto front (PF). Manuscript received September 26, 2016; revised February 25, 2017, May 31, 2017, and July 26, 2017; accepted August 15, 2017. Date of publication August 25, 2017; date of current version July 27, 2018. This work was sup- ported in part by the National Natural Science Foundation of China under Grant 61300159, Grant 61732006, Grant 61473241, and Grant 61332002, in part by the Natural Science Foundation of Jiangsu Province of China under Grant BK20130808, in part by the China Post-Doctoral Science Foundation under Grant 2015M571751, and in part by the Grant from ANR/RCC Joint Research Scheme sponsored by the Research Grants Council of the Hong Kong Special Administrative Region, China and France National Research Agency under Project A-CityU101/16. (Corresponding author: Xinye Cai.) X. Cai and Z. Mei are with the College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China, and also with the Collaborative Innovation Center of Novel Software Technology and Industrialization, Nanjing 210023, China (e-mail: xinye@nuaa.edu.cn; zwmei@nuaa.edu.cn). Z. Fan is with the Guangdong Provincial Key Laboratory of Digital Signal and Image Processing and the Department of Electronic Engineering, School of Engineering, Shantou University, Shantou 515063, China (e-mail: zfan@stu.edu.cn). Q. Zhang is with the Department of Computer Science, City University of Hong Kong, Hong Kong (e-mail: qingfu.zhang@cityu.edu.hk). This paper has supplementary downloadable material available at http://ieeexplore.ieee.org provided by the author. This consists of a PDF file containing additional material not included in the paper itself. This material is 13.4 MB in size. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEVC.2017.2744674 I. I NTRODUCTION A LONG with domination-based (e.g., [7], [12], [23], [24], [28], [40], and [48]) and indicator-based (e.g., [2]–[4], [20], and [47]) multiobjective evolution- ary algorithms (MOEAs), decomposition-based MOEAs (e.g., [13], [18], [19], [31], [32], [34], and [42]) have been recognized as a major type of approaches to tackle multiob- jective optimization problems (MOPs). As a representative of such approaches, the MOEA based on decomposition (MOEA/D) [42] has drawn a large amount of attention over the recent years. One critical difficulty for MOEA/D is on how to approximate a set of uniformly distributed Pareto optimal solutions without knowing the shape of the Pareto front (PF) [9], [30] a priori. Commonly used decomposition approaches in MOEA/D including weighted sum (WS), Tchebycheff (TCH), and penalty-based boundary intersection (PBI) [30] may fail to achieve such a goal due to the following two reasons [29], [37]. First, WS, TCH, and PBI tend to be very sensitive to the shapes of PFs [42]. An example of the Pareto optimal solu- tions obtained by TCH on MOPs with an extremely convex or concave PF is given in Fig. 1(a). Although the Pareto optimal solutions obtained by TCH are well-distributed on the concave PF, the distribution of the solutions on the extremely convex PF is not satisfactory. In Fig. 1(b), another example shows the Pareto optimal solutions obtained by TCH on MOPs with disparately scaled objectives. It can be clearly seen that these Pareto optimal solutions are very unevenly distributed on PF, where almost half of the PF is not covered by any Pareto optimal solution. It is worth noting that there has already been some research in the literature to address either one of the above scenarios. As far as we know, an inverted PBI has been proposed to tackle MOPs with extremely convex PFs in [33]. However, the use of inverted PBI to achieve well-distributed solution set still needs to assume the convexity of PFs. A combination of normal boundary intersection and the TCH approach has been proposed for MOPs with disparately scaled objectives in [43], where a satisfactory distribution of solutions can be achieved in bi-objective optimization problems but fails to extend to tri-objective optimization problems. Second, in those commonly used decomposition methods, the same solution is very likely to be assigned to many different subproblems, which may lead to the loss of diver- sity [29], [37]. The reason of such phenomenon can be explained as follows. Let x i be the current solution for the ith subproblem, then the improvement region of a solution x i 1089-778X c  2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.