Evolutionary Multitasking With Global and Local Auxiliary Tasks for Constrained Multi-Objective Optimization Kangjia Qiao, Jing Liang, Senior Member, IEEE, Zhongyao Liu, Kunjie Yu, Caitong Yue, and Boyang Qu Abstract—Constrained multi-objective optimization problems (CMOPs) include the optimization of objective functions and the satisfaction of constraint conditions, which challenge the solvers. To solve CMOPs, constrained multi-objective evolutionary algo- rithms (CMOEAs) have been developed. However, most of them tend to converge into local areas due to the loss of diversity. Evo- lutionary multitasking (EMT) is new model of solving complex optimization problems, through the knowledge transfer between the source task and other related tasks. Inspired by EMT, this paper develops a new EMT-based CMOEA to solve CMOPs, in which the main task, a global auxiliary task, and a local auxiliary task are created and optimized by one specific population respec- tively. The main task focuses on finding the feasible Pareto front (PF), and global and local auxiliary tasks are used to respectively enhance global and local diversity. Moreover, the global auxiliary task is used to implement the global search by ignoring con- straints, so as to help the population of the main task pass through infeasible obstacles. The local auxiliary task is used to provide local diversity around the population of the main task, so as to exploit promising regions. Through the knowledge transfer among the three tasks, the search ability of the population of the main task will be significantly improved. Compared with other state-of-the-art CMOEAs, the experimental results on three benchmark test suites demonstrate the superior or competitive performance of the proposed CMOEA. Index Terms—Constrained multi-objective optimization, evolu- tionary multitasking (EMT), global auxiliary task, knowledge trans- fer, local auxiliary task.    I. Introduction C ONSTRAINED multi-objective optimization problems (CMOPs) widely exist in scientific research and practical applications [1]–[6], and a CMOP can be expressed by min f ( x) = ( f 1 ( x), f 2 ( x),..., f M ( x)) (1) s.t.      g j ( x) ≤ 0, j = 1,..., p h j ( x) = 0, j = p + 1,..., n (2) f ( x) x g j ( x) h j ( x) jth ( j - p)th where indicates M optimization objectives; is the deci- sion vector; and and are the inequality con- straint and the equality constraint respectively. The violation degree of each constraint is calculated by CV j ( x) =        max ( 0, g j ( x) ) , j = 1,..., p max ( 0,   h j ( x)    - δ ) , j = p + 1,..., n (3) where δ is a positive number to relax the equality constraint. Furthermore, the total violation degree is calculated by CV ( x) = n ∑ j=1 CV j ( x) (4) CV ( x) x where = 0 indicates that is a feasible solution. ϵ Solving a CMOP involves finding a set of feasible Pareto optimal solutions, in which the mapping vectors in the objec- tive space are called feasible Pareto front (PF) or constrained Pareto front (CPF). Generally, CMOPs contain discontinuous feasible regions and nondifferentiable objective functions; thus, classical optimization methods [7]–[9] are incapable of effectively solving a majority of CMOPs. Conversely, con- strained multi-objective evolutionary algorithms (CMOEAs) have obtained significant progress in addressing complicated CMOPs due to its population-based search mechanism that can save a set of high-quality solutions [10], [11]. In addition, they do not rely on gradient information. Moreover, con- strained handling techniques, such as, the penalty function methods [12], constrained dominance principle [13], and methods [14], can be embedded in CMOEAs to help the popu- lation approach feasible regions gradually. During the past two decades, some CMOEAs have been proposed to address CMOPs. However, to guarantee feasibil- ity, constraints are often given higher priority than objectives, so the population will lose diversity and some optimal feasi- ble regions. To address these issues, some multi-stage and multi-population CMOEAs are proposed, and they begin to Manuscript received October 22, 2022; accepted November 9, 2022. This work was supported in part by the National Natural Science Fund for Outstanding Young Scholars of China (61922072), the National Natural Science Foundation of China (62176238, 61806179, 61876169, 61976237), China Postdoctoral Science Foundation (2020M682347), the Training Program of Young Backbone Teachers in Colleges and Universities in Henan Province (2020GGJS006), and Henan Provincial Young Talents Lifting Project (2021HYTP007). Recommended by Associate Editor Tao Yang. (Corresponding author: Jing Liang.) Citation: K. J. Qiao, J. Liang, Z. Y. Liu, K. J. Yu, C. T. Yue, and B. Y. Qu, “Evolutionary multitasking with global and local auxiliary tasks for constrained multi-objective optimization,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 10, pp. 1951–1964, Oct. 2023. K. J. Qiao, J. Liang, K. J. Yu, and C. T. Yue are with the School of Electrical and Information Engineering, Zhengzhou University, Zhengzhou 450001, China (e-mail: qiaokangjia@yeah.net; liangjing@zzu.edu.cn; yukunjie @zzu.edu.cn; zzuyuecaitong@163.com). Z. Y. Liu is with the School of Mechanical and Power Engineering, Zhengzhou University, Zhengzhou 450001, China (e-mail: zhongyaoliu@hot- mail.com). B. Y. Qu is with the School of Electrical and Information, Zhongyuan University of Technology, Zhengzhou 450007, China (e-mail: qby1984@hot- mail.com). 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/JAS.2023.123336 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 10, NO. 10, OCTOBER 2023 1951