Large-scale Many-objective Optimization Research Articles

Many-Objective Whale Optimization Algorithm for Engineering Design and Large-Scale Many-Objective Optimization Problems

In this paper, a novel Many-Objective Whale Optimization Algorithm (MaOWOA) is proposed to overcome the challenges of large-scale many-objective optimization problems (LSMOPs) encountered in diverse fields such as engineering. Existing algorithms suffer from curse of dimensionality i.e., they are unable to balance convergence with diversity in extensive decision-making scenarios. MaOWOA introduces strategies to accelerate convergence, balance convergence and diversity in solutions and enhance diversity in high-dimensional spaces. The prime contributions of this paper are—development of MaOWOA, incorporation an Information Feedback Mechanism (IFM) for rapid convergence, a Reference Point-based Selection (RPS) to balance convergence and diversity and a Niche Preservation Strategy (NPS) to improve diversity and prevent overcrowding. A comprehensive evaluation demonstrates MaOWOA superior performance over existing algorithms (MaOPSO, MOEA/DD, MaOABC, NSGA-III) across LSMOP1-LSMOP9 benchmarks and RWMaOP1-RWMaOP5 problems. Results validated using Wilcoxon rank sum tests, highlight MaOWOA excellence in key metrics such as generational distance, spread, spacing, runtime, inverse generational distance and hypervolume, outperforming in 71.8% of tested scenarios. Thus, MaOWOA represents a significant advancement in many-objective optimization, offering new avenues for addressing LSMOPs and RWMaOPs’ inherent challenges. This paper details MaOWOA development, theoretical basis and effectiveness, marking a promising direction for future research in optimization strategies amidst growing problem complexity.

Multipopulation-Based Differential Evolution for Large-Scale Many-Objective Optimization.

In recent years, numerous efficient many-objective optimization evolutionary algorithms have been proposed to find well-converged and well-distributed nondominated optimal solutions. However, their scalability performance may deteriorate drastically to solve large-scale many-objective optimization problems (LSMaOPs). Encountering high-dimensional solution space with more than 100 decision variables, some of them may lose diversity and trap into local optima, while others may achieve poor convergence performance. This article proposes a multipopulation-based differential evolution algorithm, called LSMaODE, which can solve LSMaOPs efficiently and effectively. In order to exploit and explore the exponential decision space, the proposed algorithm divides the population into two groups of subpopulations, which are optimized with different strategies. First, the randomized coordinate descent technique is applied to 10% of individuals to exploit the decision variables independently. This subpopulation maintains diversity in the decision space to avoid premature convergence into local optimum. Second, the remaining 90% of individuals are optimized with the nondominated guided random interpolation strategy, which interpolates individual among three nondominated solutions randomly. The strategy can guide the population convergent toward the nondominated solutions quickly, meanwhile, maintain good distribution in objective space. Finally, the proposed LSMaODE is evaluated on the LSMOP test suites from the scalability in both decision and objective dimensions. The performance is compared against five state-of-the-art large-scale many-objective evolutionary algorithms. The experimental results show that LSMaODE provides highly competitive performance.

A universal large-scale many-objective optimization framework based on cultural learning

When solving large-scale many-objective optimization problems (LMaOPs), due to the large number of variables and objectives involved, the algorithm is faced with a very high-dimensional and complex search space, which is difficult to be explored with limited resources. To address these issues, this paper proposes a universal large-scale many-objective optimization framework based on cultural learning (UCLMO). First, a universal framework is proposed, and multi-objective optimizers can be embedded into the framework to accelerate the convergence. Moreover, inspired by cultural learning, an individual selection strategy based on historical knowledge is proposed to promote the diversity of the population, and an assisted evolution strategy based on normative knowledge is presented to accelerate the convergence of the algorithm. Experiments have been conducted on multi-objective knapsack problems and LMaOPs with decision variables ranging from 500 to 1500, and the number of objectives ranging from 5 to 15. The experimental results verify the superiority and competitiveness of the proposed UCLMO framework in solving LMaOPs compared with state-of-the-art algorithms.

Solution of Large-Scale Many-Objective Optimization Problems Based on Dimension Reduction and Solving Knowledge-Guided Evolutionary Algorithm

There are lots of many-objective optimization problems (MaOPs) in real-world applications, which often have many decision variables. Although a variety of methods have been proposed to solve MaOPs, with the increasing number of decision variables or objective functions, the performance of these algorithms deteriorates appreciably. In view of this, the paper proposes a method to solve large-scale MaOPs (LSMaOPs) based on dimension reduction and a solving knowledge guided evolutionary algorithm. First, a dimension reduction method of objective functions is proposed. By clustering and aggregating the objective functions based on their correlation, the dimension of the original LSMaOP is effectively reduced. In addition, the correlations between the reduced objective functions are relatively low, so they can better represent different preferences. Then, we propose a solving knowledge guided evolutionary algorithm to solve the transformed LSMaOP. In order to get a better set of initial solutions, a population initialization method by mirror partitioning the decision space is given, in which we dynamically modify the sampling probability according to the performance of solutions contained in each sub-domain. At the same time, the algorithm will continuously supplement new excellent individuals using the solving knowledge obtained in the evolution of the population. To examine the performance of the proposed method, we carried out a number of comparative experiments. The experimental results demonstrated that the proposed algorithm can effectively tackle LSMaOPs.

Objective Space-Based Population Generation to Accelerate Evolutionary Algorithms for Large-Scale Many-Objective Optimization

The generation and updating of solutions, e.g., crossover and mutation, of many existing evolutionary algorithms directly operate on decision variables. The operators are very time consuming for large-scale and many-objective optimization problems. Different from them, this work proposes an objective space-based population generation method to obtain new individuals in the objective space and then map them to decision variable space and synthesize new solutions. It introduces three new objective vector generation methods and uses a linear mapping method to tightly connect objective space and decision one to jointly determine new-generation solutions. A loop can be formed directly between two spaces, which can generate new solutions faster and use more feedback information in the objective space. In order to demonstrate the performance of the proposed algorithm, this work performs a series of empirical experiments involving both large-scale decision variables and many objectives. Compared with the state-of-the-art traditional and large-scale algorithms, the proposed method exceeds or at least reaches its peers’ best level in overall performance while achieving great saving in execution time.

An enhancing many-objective evolutionary algorithm using chaotic mapping and solution ranking mechanism for large-scale optimization

AbstractThere are many complex optimization problems in the real world, and various evolutionary algorithms are proposed to solve them. Recently, the many-objective evolutionary algorithm using a one-by-one selection strategy (1by1EA) adopts a convergence indicator and a distribution indicator to balance convergence and diversity. However, the algorithm is too random in initialization and the fitness evaluation of solutions in the mating selection is single, which leads to poor performance in solving large-scale problems. Therefore, this paper proposes an improved method called 1by1EA-CHV by using circle chaotic mapping and a solution ranking mechanism based on the hypervolume (HV) indicator. We first map each component of solutions into a certain value space to initialize the population. Then, we calculate the contribution of each partition divided based on HV and apply the aggregation method to guide the reallocation of fitness, which achieves the ranking of solutions by using it before the old calculation method. To validate the performance, experiments compared 1by1EA-CHV with 1by1EA and other seven many-objective algorithms on large-scale functions, and the differences between these algorithms were analyzed statistically by a non-parametric test. The results showed the superiority of 1by1EA-CHV in solving large-scale many-objective optimization problems with up to 2000 decision variables.

Improving evolutionary algorithms with information feedback model for large-scale many-objective optimization

Improving evolutionary algorithms with information feedback model for large-scale many-objective optimization

Recommendation Based on Large-Scale Many-Objective Optimization for the Intelligent Internet of Things System

Recommender systems are of great significance for mining the data generated by the Internet of Things (IoT) and are important for the intelligent IoT systems. The traditional recommendation algorithms only consider the accuracy as the optimization objective. In this article, a many-objective optimization model consisting of the F1 measure, recommendation novelty, recommendation coverage, customer satisfaction, landmark similarity, and overfitting is constructed for recommendation. Then, to improve the recommendation performance, we propose to use a large-scale many-objective optimization algorithm based on problem transformation (LSMaOA) to optimize the matrix factorization model for the recommender system in the intelligent IoT systems. The experimental results show that LSMaOA is robust and can effectively optimize the model’s six objectives. Compared with the knee point-driven evolutionary algorithm (KnEA), the grid-based evolutionary algorithm (GrEA), the large-scale multiobjective optimization framework (LSMOF), and the reference vector guided evolutionary algorithm (RVEA), the proposed algorithm can promote the F1 measure by 7.78%, 13.63%, 21.85%, and 28.63%, respectively.

Solve large‐scale many‐objective optimization problems based on dual analysis of objective space and decision space

A method based on decision variable partition is often utilized for large-scale optimization problems, but it only considers the convergence and diversity of decision variables. However, decision variables in the same group may still have different (strong or weak) correlations with objectives, even if they are only related to convergence or diversity. This situation not only reduces the accuracy of the grouping of decision variables, but also affects the allocation of computational resources. To address this issue, an algorithm based on dual analysis of objective space and decision space to group decision variables more accurately for solving large-scale many-objective optimization problems is proposed; this algorithm is called LM-DAS. It first divides decision variables into two categories: convergence-related variables and the diversity-related variables. Then, the correlation between convergence-related variables and objective functions is analyzed. The decision space is divided into several subspaces according to the objective importance. Next, the decision variables are grouped using the method of interaction analysis between two level variables. Under the premise of ensuring the correctness of the grouping as much as possible, the algorithm can allocate more search resources to decision variables that are related to more important objective functions. The performance of LM-DAS is compared with four representative algorithms for 71 test instances. These instances involved 5–10 objectives and 200–1000 decision variables. The experimental results demonstrate that the proposed algorithm is competitive and effective.

A memetic evolution system with statistical variable classification for large-scale many-objective optimization

Although the optimization algorithms have been widely studied, the large-scale many-objective optimization problems (LSMaOPs) remain challenging. Due to the existence of a large number of decision variables, it is necessary to carry out decision variable analysis. However, it is often difficult to discriminate the diversity-related and convergence-related variables when the problem has complex characteristics. Meanwhile, as the number of decision variables and the number of objectives increase, many algorithms will suffer from the convergence challenge. To overcome these challenges, this paper proposes a Memetic Evolution System with Statistical Variable Classification (MES-SVC). A statistical variable classification method is proposed to discriminate the convergence-related and the diversity-related variables. A memetic evolution system, which includes a memetic exploitation and exploration module, and a memetic elite imitation module, is proposed to make information guidance during the evolution, thereby promote convergence. The performance of MES-SVC is compared with the state-of-the-art algorithms on 50 test instances with 3 to 10 objectives and 300 to 1000 decision variables. Experimental studies demonstrate the promising performance of the proposed MES-SVC in terms of both diversity and convergence of solutions.

Crashworthiness design and optimization of nested structures with a circumferentially corrugated circular outer wall and inner ribs

Twenty-six nested structures with a circumferentially corrugated circular outer wall and inner ribs are investigated under quasi-static axial compression using the finite element method. The finite element model is validated by literature. Also, theoretical predictions for the nested structures are derived and compared with finite element analyses. Among the structures whose crashworthiness performances are examined, CO1D and CSIN2D are selected as the best ones. The comparisons have shown that the ribs attached to the closest points of the corrugated outer wall provide stable collapse modes and better crash performance. Also, the crashworthiness performance of the nested structures with octagon inner structures is higher than other alternative structures because the number of corner members and the cross-section lengths are more efficient. Finally, the objective functions developed by the surrogate modeling method are optimized by non-dominated sorting genetic algorithm II (NSGA-II), multi-objective particle swarm optimization (MOPSO), paired offspring generation for constrained large-scale multiobjective optimization (POCEA) and an evolutionary algorithm for large-scale many-objective optimization (LMEA). Optimum CO1D and CSIN2D designs obtained by the NSGAII method have 37.00% and 26.68% higher SEA values, respectively than ‘Cribs’ structure at the same PCF value.

Tensor decomposition-based alternate sub-population evolution for large-scale many-objective optimization

• Introduce tensor decomposition to divide higher-dimensional heterogeneous decision variables. • The alternate sub-population evolution asymptotically optimizes the whole population. • A cross-population matching scheme is designed to reconstruct the whole population accurately. • Tensor decomposition-based model can be applied to other MOEAs with large-scale variables. A novel alternate evolution of sub-populations based on tensor decomposition called, TASE is proposed, for solving multi- and many-objective optimization problems with large-scale decision variables in this work. Tensor canonical polyadic (CP) decomposition is introduced for the first time to divide the heterogeneous variables of higher-dimensional decision space into several lower-dimensional sub-components. Furthermore, these sub-populations are optimized alternatively to search for improved solutions in their respective lower-dimensional decision subspace . Finally, a cross-population matching scheme is designed to reconstruct the whole population accurately. The experiments use some largescale multi- and many-objective problems with 2–6 objectives and 1000–5000 variables. The proposed algorithm is compared with other state-of-the-art algorithms, and the experimental results indicate that it can solve some problems that other well-known large-scale optimization algorithms cannot, as well as outperforming other algorithms in terms of solution quality and convergence rate.

Two-stage multi-tasking transform framework for large-scale many-objective optimization problems

Real-world optimization applications in complex systems always contain multiple factors to be optimized, which can be formulated as multi-objective optimization problems. These problems have been solved by many evolutionary algorithms like MOEA/D, NSGA-III, and KnEA. However, when the numbers of decision variables and objectives increase, the computation costs of those mentioned algorithms will be unaffordable. To reduce such high computation cost on large-scale many-objective optimization problems, we proposed a two-stage framework. The first stage of the proposed algorithm combines with a multi-tasking optimization strategy and a bi-directional search strategy, where the original problem is reformulated as a multi-tasking optimization problem in the decision space to enhance the convergence. To improve the diversity, in the second stage, the proposed algorithm applies multi-tasking optimization to a number of sub-problems based on reference points in the objective space. In this paper, to show the effectiveness of the proposed algorithm, we test the algorithm on the DTLZ and LSMOP problems and compare it with existing algorithms, and it outperforms other compared algorithms in most cases and shows disadvantage on both convergence and diversity.

A comparative study of many-objective optimizers on large-scale many-objective software clustering problems

Over the past 2 decades, several multi-objective optimizers (MOOs) have been proposed to address the different aspects of multi-objective optimization problems (MOPs). Unfortunately, it has been observed that many of MOOs experiences performance degradation when applied over MOPs having a large number of decision variables and objective functions. Specially, the performance of MOOs rapidly decreases when the number of decision variables and objective functions increases by more than a hundred and three, respectively. To address the challenges caused by such special case of MOPs, some large-scale multi-objective optimization optimizers (L-MuOOs) and large-scale many-objective optimization optimizers (L-MaOOs) have been developed in the literature. Even after vast development in the direction of L-MuOOs and L-MaOOs, the supremacy of these optimizers has not been tested on real-world optimization problems containing a large number of decision variables and objectives such as large-scale many-objective software clustering problems (L-MaSCPs). In this study, the performance of nine L-MuOOs and L-MaOOs (i.e., S3-CMA-ES, LMOSCO, LSMOF, LMEA, IDMOPSO, ADC-MaOO, NSGA-III, H-RVEA, and DREA) is evaluated and compared over five L-MaSCPs in terms of IGD, Hypervolume, and MQ metrics. The experimentation results show that the S3-CMA-ES and LMOSCO perform better compared to the LSMOF, LMEA, IDMOPSO, ADC-MaOO, NSGA-III, H-RVEA, and DREA in most of the cases. The LSMOF, LMEA, IDMOPSO, ADC-MaOO, NSGA-III, and DREA, are the average performer, and H-RVEA is the worst performer.

Study on Large-Scale Many-Objective Optimization Method for Aerodynamics and Underhood Cooling Airflow Using Detailed Vehicle Model

Study on Large-Scale Many-Objective Optimization Method for Aerodynamics and Underhood Cooling Airflow Using Detailed Vehicle Model

Enhancing MOEA/D with information feedback models for large-scale many-objective optimization

A multi-objective evolutionary algorithm based on decomposition (MOEA/D) is a classic decomposition-based multi-objective optimization algorithm. In the standard MOEA/D algorithm, the update process of individuals is a forward search process without using the information of previous individuals. However, there is a lot of useful information in the previous iteration. Information Feedback Models (IFM) is a new strategy which can incorporate the information from previous iteration into the updating process. Therefore, this paper proposes a MOEA/D algorithm based on information feedback model, called MOEA/D-IFM. According to the different information feedback models, this paper proposes six variants of MOEA/D, and these algorithms can be divided into two categories according to the way of selecting individuals whether it is random or fixed. At the same time, a new selection strategy has been introduced to further improve the performance of MOEA/D-IFM. The experiments were carried out in four aspects. MOEA/D-IFM were compared with other state-of-the-art multi-objective evolutionary algorithms using CEC 2018 problems in two aspects. The best one of the six improved algorithms was chosen to test on large-scale many-objective problems. In addition, we also use MOEA/D-IFM to solve multi-objective backpack problems.

Improving NSGA-III algorithms with information feedback models for large-scale many-objective optimization

Recently, more and more multi/many-objective algorithms have been proposed. However, most evolutionary algorithms only focus on solving small-scale multi/many-objective optimization problems, and few researchers pay attention to solving large-scale optimization problems. Like NSGA-III, although it performs well in solving many-objective optimization problems, it does not perform well in large-scale optimization problems. To solve this problem, in this paper, we introduce the information feedback models to improve the ability of NSGA-III to solve large-scale optimization problems. In the feedback models, the historical information of individuals in the previous generations is used in the update process of the current generation. According to the different ways of selecting individual, information feedback models are divided into two categories: one is to select individuals in a fixed way (called M-F) and the other is to select individuals randomly (called M-R). For each type of model, if the number of individuals selected is different, the information feedback models also have some difference. According to the number and mode of individual selection, there are six models in this paper. These six models are respectively embedded into NSGA-III to generate six improving NSGA-III algorithms (these six algorithms are collectively referred to as IFM-NSGAIII). These six algorithms are compared with the original NSGA-III on 9 benchmark problems to search the best information feedback model and the best IFM-NSGAIII algorithm. Then the two best IFM-NSGAIII algorithms are compared with four state-of-the-art algorithms on 9 test functions. Experiments show that the proposed algorithms are highly competitive on test problems.

Solving large-scale many-objective optimization problems by covariance matrix adaptation evolution strategy with scalable small subpopulations

Despite the recent development in evolutionary multi- and many-objective optimization, the problems with large-scale decision variables still remain challenging. In this work, we propose a scalable small subpopulations based covariance matrix adaptation evolution strategy, namely S3-CMA-ES, for solving many-objective optimization problems with large-scale decision variables. The proposed S3-CMA-ES attempts to approximate the set of Pareto-optimal solutions using a series of small subpopulations instead of a whole population, where each subpopulation converges to only one solution. In the proposed S3-CMA-ES, a diversity improvement strategy is designed to generate and select new solutions. The performance of S3-CMA-ES is compared with five representative algorithms on 36 test instances with 5–15 objectives and 500–1500 decision variables. The empirical results demonstrate the superiority of the proposed S3-CMA-ES.

A Decision Variable Clustering-Based Evolutionary Algorithm for Large-Scale Many-Objective Optimization

The current literature of evolutionary many-objective optimization is merely focused on the scalability to the number of objectives, while little work has considered the scalability to the number of decision variables. Nevertheless, many real-world problems can involve both many objectives and large-scale decision variables. To tackle such large-scale many-objective optimization problems (MaOPs), this paper proposes a specially tailored evolutionary algorithm based on a decision variable clustering method. To begin with, the decision variable clustering method divides the decision variables into two types: 1) convergence-related variables and 2) diversity-related variables. Afterward, to optimize the two types of decision variables, a convergence optimization strategy and a diversity optimization strategy are adopted. In addition, a fast nondominated sorting approach is developed to further improve the computational efficiency of the proposed algorithm. To assess the performance of the proposed algorithm, empirical experiments have been conducted on a variety of large-scale MaOPs with up to ten objectives and 5000 decision variables. Our experimental results demonstrate that the proposed algorithm has significant advantages over several state-of-the-art evolutionary algorithms in terms of the scalability to decision variables on MaOPs.