In recent years, many multimodal multi-objective evolutionary algorithms (MMOEAs) have been proposed for the widely existing multimodal multi-objective optimization problems (MMOPs) in real-world applications. However, these methods, developed based on artificial benchmark problems with a small number of decision variables, ignore the time efficiency and are ineffective for real-world MMOPs with high-dimensional decision space. To address this issue, this work proposes a time-efficient MMOEA based on SPEA2 by modifying its diversity measure and solution truncation strategy to reduce time complexity to enable solving MMOPs in limited time. Moreover, unlike the common practice in experiments for assessing MMOEAs, we set a limit on the run time rather than sufficient function evaluations as the termination condition. The results show that our algorithm obtains competitive performance on benchmark MMOPs but significantly better performances on real-world applications, including map-based location and feature selection problems, over representative and advanced MMOEAs.

Learning to optimize and automated algorithm design are attracting increasing attention, but it is still in its infancy in constrained multiobjective optimization evolutionary algorithms (CMOEAs). Current learning-assisted CMOEAs are typically crafted by human experts using manually designed techniques, which tend to be overly tuned, ad hoc, and lacking versatility. To alleviate these limitations, this work proposes transforming the online configuration of CMOEA into determinations of discrete and continuous parameters, which are then solved by deep reinforcement learning (DRL) techniques. Specifically, the Actor-Critic framework is adapted to determine a factor that defines the environmental selection pressure. The deep Q-learning technique is adopted to determine the operators for producing offspring. Owing to the property of DRL, the configured algorithm can accommodate historical experience, current evolutionary dynamics, and future improvements to achieve self-learning. A new CMOEA is proposed using the automatically configured evolutionary algorithm. Experiments on four challenging benchmarks and 21 real-world problems verify that our method significantly outperforms 11 state-of-the-art methods. The versatility and superiority of the automatically configured environment and operators over handcrafted methods justify the effectiveness of the automated configuration method, demonstrating a promising direction in evolutionary multiobjective optimization.

Constrained multi-objective optimization problems (CMOPs) have multiple objective functions that need to be optimized and constraints need to be satisfied, making them difficult to solve. Based on the multitasking optimization, the optimization of the original CMOP can be transformed into multiple related sub-tasks. Existing multitasking-based constrained multi-objective optimization evolutionary algorithms assist the evolution of the original problem by adopting auxiliary tasks. However, this approach may waste computational resources on tasks that are unsuitable for evolutionary states and dynamics. In this paper, a new competitive multitasking-based framework is proposed for CMOPs. We maintain an archive for the constrained Pareto front and multiple sub-tasks as auxiliaries. In each iteration, one of the sub-tasks is selected as the main task, and offspring are generated from its evolution. The offspring are viewed as knowledge and fed back to auxiliary tasks. The reward is mapped to a selection probability to control the main task selection in each iteration. Computational resources are saved by allocating only to the main task that is better suited for different evolutionary stages of different problems. The effectiveness of our approach is validated through experiments on four CMOP benchmark suites compared to eleven state-of-the-art methods.

The decomposable feature of operations in the welding shop scheduling scenario results in a vast search space, posing challenges for the design of traditional optimization algorithms. Addressing the multiobjective distributed heterogeneous welding shop scheduling problem (DHWSP), this work introduces a generalized multitasking framework. It establishes an auxiliary task by employing knowledge-and-learning-synergy neighborhood search, thereby enhancing the convergence and diversity of the original task. In this framework, an enhanced competitive swarm optimizer is adopted as the original task for DHWSP. Additionally, knowledge expression and transfer strategies are designed to expedite the comprehensive performance of each task by leveraging knowledge gained from search results. Finally, a memetic algorithm based on the multitasking framework is proposed for DHWSP. The effectiveness of the algorithm is validated through extensive experiments on 20 DHWSP instances. Numerical experimental results indicate that the proposed multitasking framework can significantly improve algorithmic comprehensive performance, demonstrating its efficacy in addressing the multiobjective DHWSP within a complex search space.

Achieving a tradeoff between convergence, feasibility, and diversity is critical for solving constrained multiobjective optimization problems (CMOPs). Existing constrained multiobjective evolutionary algorithms (CMOEAs) primarily focus on constraint-handling techniques to balance constraint satisfaction and objective optimization. However, individual diversity is generally considered to be low. Owing to the insufficient enhancement of diversity, CMOEAs are unable to disperse well in the objective space to enhance the search for the constrained Pareto front (CPF) when handling CMOPs with complex constraints. To address this limitation, this study develops a diversity-enhanced tri-stage framework with three different evolutionary stages. First, sufficient convergence is enabled to move the population across the infeasible regions. Afterward, an angle-domination strategy is designed, aiming to spread the population evenly in the objective space while maintaining the achieved convergence. Third, we propose a minimum neighborhood-based domination strategy to ensure that the population searches the CPF by pursuing an even distribution in the objective space. Moreover, a weight vector preselection strategy is proposed to reduce computational overhead by avoiding ineffective searches in regions that do not include the CPF. Extensive experiments with 48 benchmark instances and 25 real-world instances validate the effectiveness of our approach over nine state-of-the-art methods.

Solving constrained multiobjective optimization problems (CMOPs) with various features and challenges via evolutionary algorithms is very popular. Existing methods usually adopt an additional helper problem to simplify and solve them by divide and conquer. This article proposes a new multitasking framework for CMOPs, borrowing the idea of evolutionary multitasking optimization. The main contributions are: 1) a multitasking framework is proposed, where a CMOP is modeled as a multitasking optimization problem with three tasks. Then, it is solved by constraint-first, constraint-ignored, and constraint-relaxed multiobjective evolutionary algorithms; 2) a knowledge expression and a transfer strategy are devised to transfer the knowledge among the tasks; and 3) based on the proposed framework, a new two-stage algorithm is presented to solve CMOPs. The effectiveness of our approach is validated through experiments on four CMOP benchmark suites and 19 real-world CMOPs.

Unlike the considerable research on solving many-objective optimization problems (MaOPs) with evolutionary algorithms (EAs), there has been much less research on constrained MaOPs (CMaOPs). Generally, to effectively solve CMaOPs, an algorithm needs to balance feasibility, convergence, and diversity simultaneously. It is essential for handling CMaOPs yet most of the existing research encounters difficulties. This article proposes a novel constrained many-objective optimization EA with enhanced mating and environmental selections, namely, CMME. It can be featured as: 1) two novel ranking strategies are proposed and used in the mating and environmental selections to enrich feasibility, diversity, and convergence; 2) a novel individual density estimation is designed, and the crowding distance is integrated to promote diversity; and 3) the θ -dominance is used to strengthen the selection pressure on promoting both the convergence and diversity. The synergy of these components can achieve the goal of balancing feasibility, convergence, and diversity for solving CMaOPs. The proposed CMME is extensively evaluated on 13 CMaOPs and 3 real-world applications. Experimental results demonstrate the superiority and competitiveness of CMME over nine related algorithms.

Surrogate-assisted evolutionary algorithms (SAEAs) have been widely used for solving complex and computationally expensive optimization problems. However, most of the existing algorithms converge slowly in the later stage. This article proposes a novel two-stage data-driven evolutionary optimization (TS-DDEO) that meets the requirements of early exploration and later exploitation. In the first stage, a surrogate-assisted hierarchical particle swarm optimization method is used to find a promising area from the entire search space. In the second stage, we propose a best-data-driven optimization (BDDO) method with a strong exploitation ability to accelerate the optimization process. BDDO has a real-time update mechanism for the surrogate model and population and uses a predefined number of ranking-top solutions to update population and surrogates. BDDO combines three surrogate-assisted evolutionary sampling strategies: 1) surrogate-assisted differential evolution sampling; 2) surrogate-assisted local search; and 3) a surrogate-assisted full-crossover (FC) strategy which is proposed to integrate existing best genotypes in the population. Experiments and analysis have validated the effectiveness of the two-stage framework, the BDDO method, and the FC strategy. Moreover, the proposed algorithm is compared with five state-of-the-art SAEAs on high-dimensional benchmark functions. The result shows that TS-DDEO performs better both in effectiveness and robustness.

Solving multiobjective optimization problems (MOPs) through metaheuristic methods gets considerable attention. Based on the classical variation operators, several enhanced operators, as well as multiobjective optimization evolutionary algorithms, have been developed. Among these operators, the competitive swarm optimizer (CSO) exhibits promising performance. However, it encounters difficulties when tackling constrained MOPs (CMOPs) with large objective spaces or complex infeasible regions. In this article, a competitive and cooperative swarm optimizer is proposed, which contains two particle update strategies: 1) the CSO provides faster convergence speed to accelerate the approximation of the Pareto front and 2) the cooperative swarm optimizer suggests a mutual-learning strategy to enhance the ability to jump out of local feasible regions or local optima. We also present a new algorithm for CMOPs. The results on four benchmark suites with 47 instances demonstrate the superiority of our approach compared with other state-of-the-art methods. Additionally, its effectiveness on large-scale CMOPs has also been verified.

To solve the constrained many-objective optimization problems (CMaOPs), the tradeoff among convergence, diversity, and feasibility is a crucial and challenging task. This article proposes a new constraint-handling technique tailored for decomposition-based many-objective evolutionary algorithms to deal with the CMaOPs effectively. Specifically, the proposed method, namely, constrained penalty boundary intersection (CPBI), is an improved aggregation function based on the penalty boundary intersection. In CPBI, the normalized overall constraint violation (CV) is embedded to pursue feasibility. In this way, by the optimization of CPBI, convergence, diversity, and feasibility can be optimized simultaneously. Furthermore, the weight of the normalized overall CV is adjusted adaptively based on the feasible ratio of the current population. To evaluate the performance of CPBI, it is combined with three decomposition-based algorithms. Ten benchmark problems with 50 instances are chosen as the test suite. In addition, the proposed method is compared with nine advanced algorithms. Experimental results have demonstrated the promising performance of CPBI for different problems.

Multiobjective optimization evolutionary algorithms (MOEAs) have received significant achievements in recent years. However, they encounter many difficulties in dealing with many-objective optimization problems (MaOPs) due to the weak selection pressure. One possible way to improve the ability of MOEAs for these MaOPs is to balance the convergence and diversity in the high-dimensional objective space. Based on this consideration, this article proposes a novel generic two-stage (TS) framework for MaOPs. The entire evolutionary search process is divided into two stages: in the first stage, a new subregion dominance and a modified subregion density-based mating selection mainly purse the convergence and in the second stage, a novel level-based Pareto dominance cooperates with the traditional Pareto dominance that mainly promotes diversity. Integrated into NSGA-II, the TS NSGA-II, referred to as TS-NSGA-II, is proposed. To extensively evaluate the performance of our approach, 29 benchmark problems were used as the test suite. The experimental results demonstrate our approach obtained superior or competitive performance compared with eight state-of-the-art many-objective optimization evolutionary algorithms. To study its generality, the proposed TS strategy was also combined with four other advanced methods for MaOPs. The results show that it can also improve the performance of these four methods in terms of convergence and diversity.