During the past decade, research efforts have been gradually directed to the widely existing yet less noticed multimodal multi-objective optimization problems (MMOPs) in the multi-objective optimization community. Recently, researchers have begun to investigate enhancing the decision space diversity and preserving valuable dominated solutions to overcome the shortage caused by a preference for objective space convergence. However, many existing methods still have limitations, such as giving unduly high priorities to convergence and insufficient ability to enhance decision space diversity. To overcome these shortcomings, this article aims to explore a promising region (PR) and enhance the decision space diversity for handling MMOPs. Unlike traditional methods, we propose the use of non-dominated solutions to determine a limited region in the PR in the decision space, where the Pareto sets (PSs) are included, and explore this region to assist in solving MMOPs. Furthermore, we develop a novel neighbor distance measure that is more suitable for the complex geometry of PSs in the decision space than the crowding distance. Based on the above methods, we propose a novel dual-population-based coevolutionary algorithm. Experimental studies on three benchmark test suites demonstrates that our proposed methods can achieve promising performance and versatility on different MMOPs. The effectiveness of the proposed neighbor distance has also been justified through comparisons with crowding distance methods.

A chip mounter is the core equipment in the production line of the surface-mount technology, which is responsible for finishing the mount operation. It is the most complex and time-consuming stage in the production process. Therefore, it is of great significance to optimize the load balance and mounting efficiency of the chip mounter and improve the mounting efficiency of the production line. In this study, according to the specific type of chip mounter in the actual production line of a company, a maximum and minimum model is established to minimize the maximum cycle time of the chip mounter in the production line. The production efficiency of the production line can be improved by optimizing the workload scheduling of each chip mounter. On this basis, a hybrid adaptive optimization algorithm is proposed to solve the load scheduling problem of the mounter. The hybrid algorithm is a hybrid of an adaptive genetic algorithm and the improved ant colony algorithm. It combines the advantages of the two algorithms and improves their global search ability and convergence speed. The experimental results show that the proposed hybrid optimization algorithm has a good optimization effect and convergence in the load scheduling problem of chip mounters.

Nonlinear Equations (NEs), which may usually have multiple roots, are ubiquitous in diverse fields. One of the main purposes of solving NEs is to locate as many roots as possible simultaneously in a single run, however, it is a difficult and challenging task in numerical computation. In recent years, Intelligent Optimization Algorithms (IOAs) have shown to be particularly effective in solving NEs. This paper provides a comprehensive survey on IOAs that have been exploited to locate multiple roots of NEs. This paper first revisits the fundamental definition of NEs and reviews the most recent development of the transformation techniques. Then, solving NEs with IOAs is reviewed, followed by the benchmark functions and the performance comparison of several state-of-the-art algorithms. Finally, this paper points out the challenges and some possible open issues for solving NEs.