Combining sparse evolutionary operators and deep reinforcement learning for large-scale sparse multiobjective optimization problems

Benefits of sparse population sampling in multi-objective evolutionary computing for large-Scale sparse optimization problems

Recently large-scale sparse multi-objective optimization problems are increasingly concerned by researchers. Different from large-scale multi-objective optimization problems, most of the decision variables (Decs) of the large-scale sparse multi-objective optimization problems are equal to zero. Among the existing large-scale sparse evolutionary algorithms, SparseEA and SparseEA2 dynamically mask some of the real decision variables to zero by setting Mask which can accelerate convergence. When determining Mask updates, the SparseEA and SparseEA2 algorithms both use a static fitness. However, static fitness is limited by the number of iterations so that it is difficult to cover the global information. To address this issue, we design an adaptive fitness for Mask updates. Moreover, we increase the number of decision variables per flip according to the number of decision variables, and gradually decrease the flip probability as the number of iterations increases. In the case of real Decs cross, we only cross the real Decs with the same Mask as the parents. We experiment on eight benchmark problems and three real-world application problems, and the simulation results show that our algorithm is significantly more efficient than the other four currently available sparse large-scale multi-objective algorithms.

A multi-stage knowledge-guided evolutionary algorithm for large-scale sparse multi-objective optimization problems

The Pareto sets of many real-world multi-objective optimization problems in engineering and computer fields are high-dimensional but sparse. Such multi-objective optimization problems are called large-scale sparse multi-objective optimization problems. A sparse evolutionary algorithm has also been raised and verified effective on benchmark problems. However, in practical applications, it needs a large number of expensive function evaluations. Although surrogate-assisted evolutionary algorithms are common solutions to deal with expensive optimization problems within limited computation resources and especially Kriging-assisted evolutionary algorithms are widely used, they cannot cope with large-scale expensive sparse multi-objective optimization problems due to the inaccurate surrogate models on high-dimensional problems. Therefore, we first propose a feature selection operator based on non-dominated sorting to choose the non-zero decision variables in the Pareto set. Then, the dimension of the original problem is reduced and a Kriging-assisted multi-objective evolutionary algorithm is employed to solve the reformulated problem. Finally, the selected zero decision variables are added to the obtained optimal solutions as the result of the original problem. The experimental results on benchmark problems show that our proposed algorithm outperforms the existing algorithms.

A Cluster-Based Competitive Particle Swarm Optimizer with a Sparse Truncation Operator for Multi-Objective Optimization

The two-cardinality sparse constrained optimization problems include sparse optimization problems and constrained sparse optimization problems in many fields, such as signal processing, image processing, securities investment etc. A smoothing penalty function method and a smoothing objective penalty function method are studied for two-cardinality sparse constrained optimization problems respectively. Some error estimations are proved for the smoothing penalty function and the smoothing objective penalty function. Based on the smoothing penalty function and smoothing objective penalty function, two algorithms are designed to solve two-cardinality sparse constrained optimization problems and their convergence is proved respectively. Numerical results show that the two algorithms have the similar effectiveness in finding out an approximate solution for two-cardinality sparse constrained optimization problems.

The performance of most multi-objective evolutionary algorithms (MOEAs) usually degenerates when they are adopted to tackle large-scale multi-objective optimization problems (LSMOPs) involving a large number of decision variables. While LSMOPs attract increasing attention in the evolutionary computation community in recent years, a number of delicate approaches have been proposed to improve the performance of MOEAs on LSMOPs. While many real-world LSMOPs include sparse optimal solutions (i.e., most decision variables are zero) that should be found within a limited budget of function evaluations, few MOEAs have been tailored for these sparse LSMOPs and the performance of existing MOEAs on them has not been well studied. In this paper, we first have a brief review of existing MOEAs for LSMOPs, and then elaborate two customized MOEAs for solving sparse LSMOPs. Next, we select six state-of-the-art MOEAs with different search strategies to conduct an experimental study on eight sparse benchmark problems and four real-world applications. Finally, we outline some future research directions of large-scale optimization involving sparsity.KeywordsLarge-scale multi-objective optimizationSparse Pareto optimal solutionsEvolutionary algorithmReal-world applications

An enhanced competitive swarm optimizer with strongly robust sparse operator for large-scale sparse multi-objective optimization problem

Due to the high-dimensional decision variables and the sparse nature of solutions in sparse large-scale multi-objective optimization problems (SLMOPs), traditional multi-objective evolutionary algorithms (MOEAs) encounter substantial challenges. Researchers have proposed various sparse evolutionary algorithms (SEAs) to address these challenges. However, most existing SEAs focus on accurately identifying non-zero variable positions while neglecting changes in objective function values during optimization. This paper introduces a Jaccard similarity coefficient-based evolutionary algorithm (JSCEA) designed to search for sparse distributions that optimize objective values rapidly and efficiently. Leveraging the Jaccard similarity coefficient (JSC) to identify critical sparse patterns among promising solutions and propagate them to subsequent generations enhances the algorithm’s computing efficiency, particularly under limited computational resources. Experimental results on three real-world problems and eight benchmark tests demonstrate that JSCEA performs competitively on problem sizes of up to 10,000 variables.

Sparse large-scale multi-objective optimization problems (LSMOPs) widely exist in real-world applications, which have the properties of involving a large number of decision variables and sparse Pareto optimal solutions, i.e., most decision variables of these solutions are zero. In recent years, sparse LSMOPs have attracted increasing attentions in the evolutionary computation community. However, all the recently tailored algorithms for sparse LSMOPs put the sparsity detection and maintenance in the first place, where the nonzero variables can hardly be optimized sufficiently within a limited budget of function evaluations. To address this issue, this paper proposes to enhance the connection between real variables and binary variables within the two-layer encoding scheme with the assistance of variable grouping techniques. In this way, more efforts can be devoted to the real part of nonzero variables, achieving the balance between sparsity maintenance and variable optimization. According to the experimental results on eight benchmark problems and three real-world applications, the proposed algorithm is superior over existing state-of-the-art evolutionary algorithms for sparse LSMOPs.

Large-scale sparse multiobjective optimization problems (LSMOPs) are of great significance in the context of practical applications, such as critical node detection, feature selection, and pattern mining. Since many LSMOPs are pursued based on large datasets, they involve a large number of decision variables, resulting in a huge search space that is challenging to explore efficiently. To rapidly approximate sparse Pareto optimal solutions, some evolutionary algorithms have been proposed to reduce the dimensionality of LSMOPs. However, their adaptability to different LSMOPs remains limited due to their reliance on fixed dimensionality reduction schemes, which can potentially lead to local optima and inefficient utilization of function evaluations. To address this issue, a knowledge learning-based dimensionality reduction approach is proposed in this article. First, in the early stages of evolution, the impact of different dimensionality reduction schemes on the sparse distribution of the population is evaluated. Then, the multilayer perceptron is employed to learn the accumulated knowledge from the evolutionary process, thereby constructing a mapping model between the sparse features of the evolutionary process and the candidate dimensionality reduction schemes. Finally, the model recommends the best dimensionality reduction scheme in each generation, achieving a good balance between exploration and exploitation. Experimental evaluations on both benchmark and real-world LSMOPs demonstrate that an evolutionary algorithm incorporating the proposed knowledge learning-based dimensionality reduction approach outperforms most existing evolutionary algorithms.

Sparse multiobjective optimization problems are common in practical applications. Such problems are characterized by large-scale decision variables and sparse optimal solutions. General large-scale multiobjective optimization problems (LSMOPs) have been extensively studied for many years. They can be well solved by many excellent custom algorithms. However, when these algorithms are used to deal with sparse LSMOPs, they often encounter difficulties because the sparse nature of the problem is not considered. Therefore, aiming at sparse LSMOPs, an algorithm based on multiple sparse detection is proposed in this paper. The algorithm applies an adaptive sparse genetic operator that can generate sparse solutions by detecting the sparsity of individuals. To improve the deficiency of sparse detection caused by local detection, an enhanced sparse detection (ESD) strategy is proposed in this paper. The strategy uses binary coefficient vectors to integrate the masks of nondominated solutions. Essentially, the mask is globally and deeply optimized by coefficient vectors to enhance the sparsity of the solutions. In addition, the algorithm adopts an improved weighted optimization strategy to fully optimize the key nonzero variables to balance exploration and optimization. Finally, the proposed algorithm is named MOEA-ESD and is compared to the current state-of-the-art algorithm to verify its effectiveness.

Large-scale sparse multi-objective optimization problems are prevalent in numerous real-world scenarios, such as neural network training, sparse regression, pattern mining and critical node detection, where Pareto optimal solutions exhibit sparse characteristics. Ordinary large-scale multi-objective optimization algorithms implement undifferentiated update operations on all decision variables, which reduces search efficiency, so the Pareto solutions obtained by the algorithms fail to meet the sparsity requirements. SparseEA is capable of generating sparse solutions and calculating scores for each decision variable, which serves as a basis for crossover and mutation in subsequent evolutionary process. However, the scores remain unchanged in iterative process, which restricts the sparse optimization ability of the algorithm. To solve the problem, this paper proposes an evolution algorithm with the adaptive genetic operator and dynamic scoring mechanism for large-scale sparse many-objective optimization (SparseEA-AGDS). Within the evolutionary algorithm for large-scale Sparse (SparseEA) framework, the proposed adaptive genetic operator and dynamic scoring mechanism adaptively adjust the probability of cross-mutation operations based on the fluctuating non-dominated layer levels of individuals, concurrently updating the scores of decision variables to encourage superior individuals to gain additional genetic opportunities. Moreover, to augment the algorithm’s capability to handle many-objective problems, a reference point-based environmental selection strategy is incorporated. Comparative experimental results demonstrate that the SparseEA-AGDS algorithm outperforms five other algorithms in terms of convergence and diversity on the SMOP benchmark problem set with many-objective and also yields superior sparse Pareto optimal solutions.

In real-world applications, there exist a lot of multiobjective optimization problems whose Pareto-optimal solutions are sparse, that is, most variables of these solutions are 0. Generally, many sparse multiobjective optimization problems (SMOPs) contain a large number of variables, which pose grand challenges for evolutionary algorithms to find the optimal solutions efficiently. To address the curse of dimensionality, this article proposes an evolutionary algorithm for solving large-scale SMOPs, which aims to mine the sparse distribution of the Pareto-optimal solutions and, thus, considerably reduces the search space. More specifically, the proposed algorithm suggests an evolutionary pattern mining approach to detect the maximum and minimum candidate sets of the nonzero variables in the Pareto-optimal solutions, and uses them to limit the dimensions in generating offspring solutions. For further performance enhancement, a binary crossover operator and a binary mutation operator are designed to ensure the sparsity of solutions. According to the results on eight benchmark problems and four real-world problems, the proposed algorithm is superior over existing evolutionary algorithms in solving large-scale SMOPs.

A two-stage evolutionary algorithm for large-scale sparse multiobjective optimization problems