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This article suggests a multimodal multiobjective evolutionary algorithm with dual clustering in decision and objective spaces. One clustering is run in decision space to gather nearby solutions, which will classify solutions into multiple local clusters. Nondominated solutions within each local cluster are first selected to maintain local Pareto sets, and the remaining ones with good convergence in objective space are also selected, which will form a temporary population with more than ${N}$ solutions ( ${N}$ is the population size). After that, a second clustering is run in objective space for this temporary population to get ${N}$ final clusters with good diversity in objective space. Finally, a pruning process is repeatedly run on the above clusters until each cluster has only one solution, which removes the most crowded solution in decision space from the most crowded cluster in objective space each time. This way, the clustering in decision space can distinguish all Pareto sets and avoid the loss of local Pareto sets, while that in objective space can maintain diversity in objective space. When solving all the benchmark problems from the competition of multimodal multiobjective optimization in the IEEE Congress on Evolutionary Computation 2019, the experiments validate our advantages to maintain diversity in both objective and decision spaces.

Dynamic multiobjective optimization problems (DMOPs) are optimization problems with multiple conflicting optimization objectives, and these objectives change over time. Transfer learning-based approaches have been proven to be promising; however, a slow solving speed is one of the main obstacles preventing such methods from solving real-world problems. One of the reasons for the slow running speed is that low-quality individuals occupy a large amount of computing resources, and these individuals may lead to negative transfer. Combining high-quality individuals, such as knee points, with transfer learning is a feasible solution to this problem. However, the problem with this idea is that the number of high-quality individuals is often very small, so it is difficult to acquire substantial improvements using conventional transfer learning methods. In this article, we propose a knee point-based transfer learning method, called KT-DMOEA, for solving DMOPs. In the proposed method, a trend prediction model (TPM) is developed for producing the estimated knee points. Then, an imbalance transfer learning method is proposed to generate a high-quality initial population by using these estimated knee points. The advantage of this approach is that the seamless integration of a small number of high-quality individuals and the imbalance transfer learning technique can greatly improve the computational efficiency while maintaining the quality of the solution. The experimental results and performance comparisons with some chosen state-of-the-art algorithms demonstrate that the proposed design is capable of significantly improving the performance of dynamic optimization.

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Constrained multiobjective optimization problems (CMOPs) are challenging because of the difficulty in handling both multiple objectives and constraints. While some evolutionary algorithms have demonstrated high performance on most CMOPs, they exhibit bad convergence or diversity performance on CMOPs with small feasible regions. To remedy this issue, this article proposes a coevolutionary framework for constrained multiobjective optimization, which solves a complex CMOP assisted by a simple helper problem. The proposed framework evolves one population to solve the original CMOP and evolves another population to solve a helper problem derived from the original one. While the two populations are evolved by the same optimizer separately, the assistance in solving the original CMOP is achieved by sharing useful information between the two populations. In the experiments, the proposed framework is compared to several state-of-the-art algorithms tailored for CMOPs. High competitiveness of the proposed framework is demonstrated by applying it to 47 benchmark CMOPs and the vehicle routing problem with time windows.

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Hypervolume is widely used as a performance indicator in the field of evolutionary multiobjective optimization (EMO). It is used not only for performance evaluation of EMO algorithms (EMOAs) but also in indicator-based EMOAs to guide the search. Since its initial proposal in the late 1990s, a wide variety of studies have been done on various topics, including hypervolume calculation, optimal $mu $ -distribution, subset selection, hypervolume-based EMOAs, and extensions of the hypervolume indicator. However, currently there is no work to systematically survey the hypervolume indicator for these topics whereas it has been frequently used in the EMO field. This article aims to fill this gap and provide a comprehensive survey on the hypervolume indicator. We expect that this survey will help EMO researchers to understand the hypervolume indicator more deeply and thoroughly, and promote further utilization of the hypervolume indicator in the EMO field.

Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted generational distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a real-world optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties, such as Pareto compliance, immunity to dominated/duplicate solutions, and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.