In evolutionary multi-objective optimization (EMO), reproducing high-quality offspring solutions is a key factor in developing effective EMO algorithms (EMOAs). Many model-based methods have been proposed to generate high-quality offspring solutions. Almost all studies use the solutions in the current and previous generations as the training data. When using existing solutions as training data, the neural network is effective at identifying promising improvement directions for poor solutions. However, it does not perform well in determining effective improvement directions for solutions that are already good. In this paper, we propose a novel method that uses momentum moves to generate new solutions and includes the good new solutions in the training dataset. The momentum method offers a novel approach to enhancing the quality of the current solutions without incurring any additional computation costs. Our approach is demonstrated on both artificial and real-world problems, and compared with the algorithm using the traditional training dataset. Results show that including the momentum-based new solutions in the training dataset significantly improves the computational efficiency of the algorithm.

Momentum is a technique that adds the momentum moves from the earlier iterations into the current update to accelerate convergence. While the momentum technique has been widely used in single-objective optimization, its application in evolutionary multi-objective optimization (EMO) has not gained much attention. Since EMO algorithms are population-based algorithms, how to choose solutions for applying momentum becomes an important issue. Inspired by Polyak’s momentum method and Nesterov’s momentum method in single-objective optimization, we propose four different momentum methods for EMO. Our findings demonstrate that the performance of EMOAs with momentum is strongly affected by the choice of solutions to which momentum moves are applied.

In the community of evolutionary multi-objective optimization (EMO), large-scale multi-objective optimization problems (LSMOPs) with many decision variables have attracted much attention. The main difficulty of LSMOPs lies in their high-dimensional decision space, which slows down the convergence of EMO algorithms towards the Pareto front. To address this issue, many novel variation operators have been proposed to improve the efficiency of EMO algorithms. However, for both conventional EMO algorithms (e.g., NSGA-II) and recently proposed EMO algorithms (e.g., LERD), the polynomial mutation with the mutation probability 1/n, where n is the number of decision variables, is always used. For LSMOPs with a large number of decision variables, the mutation probability 1/n looks too small (e.g., 1/1000). In this paper, we examine different mutation probabilities and find that many existing EMO algorithms with a larger mutation probability (e.g., 10/n) are significantly better than the standard setting (i.e., 1/n) in handling LSMOPs.

To improve the convergence ability of evolutionary multi-objective optimization algorithms (EMOAs), various strategies have been proposed. One effective strategy is to use good momentum from the previous generations to create new solutions. However, the definition of good momentum has not been carefully studied. In this paper, we propose five different definitions of good momentum for EMOAs. Then, we explain their integration into popular EMOAs such as NSGA-II, MOEA/D, and SMS-EMOA. Through computational experiments, we demonstrate that the use of an appropriate definition of good momentum greatly accelerates the convergence of EMOAs on both artificial test problems and real-world problems, particularly on large-scale problems.

Recently, Pareto Set Learning (PSL) has been proposed for learning the entire Pareto set using a neural network. PSL employs preference vectors to scalarize multiple objectives, facilitating the learning of mappings from preference vectors to specific Pareto optimal solutions. Previous PSL methods have shown their effectiveness in solving artificial multi-objective optimization problems (MOPs) with uniform preference vector sampling. The quality of the learned Pareto set is influenced by the sampling strategy of the preference vector, and the sampling of the preference vector needs to be decided based on the Pareto front shape. However, a fixed preference sampling strategy cannot simultaneously adapt the Pareto front of multiple MOPs. To address this limitation, this paper proposes an Evolutionary Preference Sampling (EPS) strategy to efficiently sample preference vectors. Inspired by evolutionary algorithms, we consider preference sampling as an evolutionary process to generate preference vectors for neural network training. We integrate the EPS strategy into five advanced PSL methods. Extensive experiments demonstrate that our proposed method has a faster convergence speed than baseline algorithms on 7 testing problems. Our implementation is available at https://github.com/rG223/EPS.

Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto _Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms.

An important issue in hypervolume-based evolutionary multi-objective optimization (EMO) algorithms is the specification of a reference point for hypervolume calculation. However, its appropriate specification has not been carefully studied in the literature. Some recent studies have pointed out the importance and difficulty of the reference point specification. Its appropriate specification depends on problem characteristics such as the Pareto front shape and the number of objectives. In this paper, the difficulty of the reference point specification in hypervolume-based EMO algorithms is circumvented by using two reference points. Instead of using only a single reference point, we propose a new hypervolume-based EMO algorithm that can effectively utilize two reference points cooperatively. Experimental results show that the proposed algorithm has good and robust performance on a wide range of test problems. In comparison to hypervolume-based EMO algorithms with only a single reference point, the proposed algorithm can find a wider and more uniformly distributed solution set. On a recently proposed real-world problem suite, the proposed algorithm shows competitive performance in comparison to state-of-the-art algorithms.

In the evolutionary multi-objective optimization (EMO) community, an unbounded external archive has been used in some studies for evaluating the performance of EMO algorithms. Those studies show that the unbounded external archive often includes better solutions than the final population. Thus, it is likely that the search ability of an EMO algorithm can be improved by periodically updating the current population using the unbounded external archive (i.e., by periodically choosing good solutions from all the examined solutions as the current population). However, the usefulness of such a global generation update scheme has not been studied in the literature. In this paper, we examine the effect of the periodical global generation update on the performance of well-known and frequently-used EMO algorithms: NSGA-II, MOEA/D and NSGA-III. We use the PBI function with uniformly distributed weight vectors for the periodical global generation update. In our computational experiments, we obtain clearly improved results by the periodical global generation update. We also examine the effect of the frequency of the global generation update (e.g., every 20 generations) on the performance of each EMO algorithm and its run time.

The research on large-scale multi- and many-objective optimization has received increasing attention in the evolutionary multi-objective optimization (EMO) community. A number of large-scale EMO algorithms based on different strategies (e.g., divide-and-conquer, coevolution, and dimensionality reduction) have been proposed over the last decade. The performance of the large-scale EMO algorithms was empirically evaluated using several benchmark test suites, including the ZDT, DTLZ, WFG, MaF, UF and LSMOP test suites. Even though these test suites are theoretically scalable to any number of decision variables, they are not necessarily appropriate for examining the performance of large-scale EMO algorithms. In fact, among these benchmark test suites, only the LSMOP test suite is specifically designed to test the performance of large-scale EMO algorithms. In this paper, we propose a new scalable multi- and many-objective test problem for examining large-scale EMO algorithms. The proposed test problem has the following features: 1) the number of objectives and decision variables can be arbitrarily specified; 2) the interaction strength among the objectives can be adjusted by a correlation parameter. The performance of six EMO algorithms is examined on the new test problem. Our experimental results show that the proposed new test problem poses difficulties to some state-of-the-art large-scale EMO algorithms.