948 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 24, NO. 5, OCTOBER 2020 Limit-Cycle-Based Mutant Multiobjective Pigeon-Inspired Optimization Haibin Duan , Senior Member, IEEE, Mengzhen Huo , and Yuhui Shi , Fellow, IEEE Abstract—This article presents a limit-cycle-based mutant multiobjective pigeon-inspired optimization (PIO). In this algo- rithm, the limit-cycle-based mechanism is devised to consider the factors that affect the flight of pigeons to simplify the multiobjective PIO algorithm. The mutant mechanism is incorpo- rated to strengthen the exploration capability in the evolutionary process. Additionally, the application of the dual repository makes the nondominated solutions stored and selected to guide the flight of pigeons. Attributed to the limit-cycle-based mutant mechanisms, this algorithm not only obtains the faster conver- gence speed and higher accuracy but also improves its population diversity. To confirm the universal application of this algorithm, theoretical analysis of the convergence is discussed in this article. Finally, comparative experiments of our proposed algorithm and other five multiobjective methods are conducted to verify the accuracy, efficiency, and convergence stability of the proposed algorithm. Index Terms—Limit-cycle-based mechanism, multiobjective pigeon-inspired optimization (PIO), mutant mechanism, theoret- ical analysis. I. I NTRODUCTION O PTIMIZATION problems are presented to solve real- life decisions and planning situations. However, there is always more than one objective to be optimized simul- taneously in many scientific and engineering applications. These problems are termed as multiobjective optimization problems (MOPs). Due to the conflicting property of multiple objectives, a sin- gle solution which can find the optimum for all the objectives Manuscript received December 3, 2019; revised February 22, 2020; accepted March 16, 2020. Date of publication March 27, 2020; date of cur- rent version October 1, 2020. This work was supported in part by the Science and Technology Innovation 2030-Key Project of “New Generation Artificial Intelligence” under Grant 2018AAA0102303 and Grant 2018AAA0102403, in part by the National Natural Science Foundation of China under Grant 91948204, Grant 61761136008, Grant U1913602, Grant U19B2033, and Grant 91648205, and in part by the Aeronautical Foundation of China under Grant 20185851022. (Corresponding author: Haibin Duan.) Haibin Duan is with the State Key Laboratory of Virtual Reality Technology and Systems, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China, and also with Peng Cheng Laboratory, Shenzhen 518000, China (e-mail: hbduan@buaa.edu.cn). Mengzhen Huo is with the School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China, and also with the Shenyuan Honors College, Beihang University, Beijing 100083, China (e-mail: mzhuo@buaa.edu.cn). Yuhui Shi is with the Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China (e-mail: shiyh@sustech.edu.cn). Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEVC.2020.2983311 at the same time does not exist. The improvement of one objective function may result in the deterioration of others [1]. Thus, MOPs are aimed to find a set of tradeoff solutions (Pareto-optimal solutions) [2], known as Pareto set (PS) in the decision space and Pareto front (PF) in the objective space. The obtained Pareto-optimal solutions can be applied to various cases in real life. Since the number of Pareto-optimal solutions is usually far more than the desired number, lots of multiobjective algo- rithms are represented to obtain a representative and diverse set of solutions for selection [3], [4]. Due to the gener- ality and population-based searching nature, multiobjective evolutionary algorithms (MOEAs) could find the approxi- mate set of Pareto-optimal solutions effectively and efficiently in a single run [5], [6]. Therefore, MOEAs have experi- enced great development [7] and played the major role in addressing complicate MOPs in the EA field. The most excel- lent representative MOEAs include the nondominated sorting genetic algorithm (NSGA) [8], MOEA based on decompo- sition (MOEA/D) [9], the strength Pareto evolutionary algo- rithm (SPEA) [10], and their variants, such as NSGA-II [11], NSGA-III [12], [13], MOEA/D-KF [14], SPEA-II [15], and other algorithms. Many multiobjective evolutionary optimization algorithms generally have difficulties in solving large-scale multiobjective problems (LSMOPs). As the dimension of decision variables increases, the decision space usually exponentially grows and the general characteristics of the problem become more com- plicated which result in the premature convergence to local optima of MOEAs due to the degradation of population diver- sity in early stages of the optimization process [16]. Thus, nature-inspired heuristic algorithms, including evolutionary algorithms (EAs) [11], [17], artificial immune algorithms [18], [19], particle swarm optimization algorithms [20], [21], and pigeon-inspired optimization (PIO) algorithms [22] have been developed to tackle MOPs. Nevertheless, on the one hand, the performance of EAs always severely degrades when solv- ing large-scale optimization problems, like DE algorithm [16]. For another, various strategies utilized in the improvement of MOEAs could bring the disadvantages of heavy calculation burden [23]. PIO is a newly proposed bio-inspired swarm intelligence algorithm, which was invented by Duan and Qiao [22] and successfully applied for solving real-world problems in vari- ous fields [24]–[26]. The basic algorithm of PIO is inspired by the behavior of the homing pigeons. In the evolutionary process of the algorithm, the pigeons could employ different 1089-778X c  2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: BEIHANG UNIVERSITY. Downloaded on October 04,2020 at 02:10:17 UTC from IEEE Xplore. Restrictions apply.