Many-objective Optimization Problems Research Articles

A many-objective evolutionary algorithm based on interaction force and hybrid optimization mechanism

In many-objective optimization, both convergence and diversity are equally important. However, in high-dimensional spaces, traditional decomposition-based many-objective evolutionary algorithms struggle to ensure population diversity. Conversely, traditional Pareto dominance-based many-objective evolutionary algorithms face challenges in ensuring population convergence. In this paper, we propose a novel many-objective evolutionary algorithm based on interaction force and hybrid optimization mechanism (MaOEAIH) for effectively addressing the difficulty in balancing convergence and diversity. First, we use the concept of interaction force to simulate the convergence (akin to gravity) and diversity (repulsion) of the population. Subsequently, we design an optimization mechanism that combines decomposition and Pareto dominance to enhance the convergence and diversity of the population separately. Simultaneously, to eliminate dominance resistance solutions, we propose a quartile method based on boundary solutions. Additionally, Random perturbations are also introduced to certain individuals within the population to facilitate their escape from local optima. MaOEAIH is compared with some state-of-the-art algorithms on 31 well-known test problems with 3-15 objectives. The experimental results show that, compared to other algorithms, MaOEAIH not only obtains solution sets of higher quality when dealing with different types of many-objective optimization problems, but also effectively addresses key challenges including insufficient selection pressure, difficulty balancing convergence and diversity, and susceptibility to population entrapment in local optima within many-objective optimization scenarios.

TRAA: a two-risk archive algorithm for expensive many-objective optimization

AbstractMany engineering problems are essentially expensive multi-/many-objective optimization problems, and surrogate-assisted evolutionary algorithms have gained widespread attention in dealing with them. As the objective dimension increases, the error of predicting solutions based on surrogate models accumulates. Existing algorithms do not have strong selection pressure in the candidate solution obtaining and adaptive sampling stages. These make the effectiveness and area of application of the algorithms unsatisfactory. Therefore, this paper proposes a two-risk archive algorithm, which contains a strategy for mining high-risk and low-risk archives and a four-state adaptive sampling criterion. In the candidate solution mining stage, two types of Kriging models are trained, then conservative optimization models and non-conservative optimization models are constructed for model searching, followed by archive selection to obtain more reliable two-risk archives. In the adaptive sampling stage, in order to improve the performance of the algorithms, the proposed criterion considers environmental assessment, demand assessment, and sampling, where the sampling approach involves the improvement of the comprehensive performance in reliable environments, convergence and diversity in controversial environments, and surrogate model uncertainty. Experimental results on numerous benchmark problems show that the proposed algorithm is far superior to seven state-of-the-art algorithms in terms of comprehensive performance.

Can multi-period auto-portfolio systems improve returns? Evidence from Chinese and U.S. stock markets

Current portfolios often underperform due to limited utilization of stock selection and a lack of attention to multi-period trading. To address this issue, we propose an auto-portfolio system that addresses these problems by integrating multi-class stock selection with portfolio optimization based on technical indicators. For stock selection, we combine Two-dimensional Convolutional Neural Network with Long and Short-term Memory to forecast the future trends of stocks and select potentially profitable stocks for investment. We then develop two portfolio models based on two technical indicators, which automatically perform multi-period investment. We establish a many-objective optimization problem including return, Conditional Value-at-Risk, skewness, kurtosis, and cost. To solve the optimization problem, we employ Non-dominated Sorting Genetic Algorithm III. The data of Chinese and the U.S. stock markets is used for verification, and a comparative analysis is discussed. In the out-of-sample period, two proposed multi-period portfolio models outperform the other models in both single and multi period, achieving higher Sharpe ratio of 1.021 and 1.052 in China, and 1.116 and 1.236 in the U.S., respectively.

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.

A many‐objective evolutionary algorithm based on bi‐direction fusion niche dominance

SummaryAlthough some many‐objective optimization algorithms (MaOEAs) have been proposed recently, Pareto dominance‐based MaOEAs still cannot effectively balance convergence and diversity in solving many objective optimization problems (MaOPs) due to insufficient selection pressure. To address this problem, a bi‐directional fusion niche domination is proposed. This method merges the strengths of cone and parallel decomposition directions in comparing dominations for nondominance stratification within the candidate population, augmenting the selection pressure of population. Subsequently, the crowding distance is introduced as an additional selection criterion to further refine the selection of nondominated individuals within the critical layer. Lastly, a MaOEA based on bi‐directional fusion niche dominance (MaOEA/BnD) is proposed, utilizing bi‐directional fusion niche dominance and crowding distance as important components of environmental selection. The performance of MaOEA/BnD was compared with five representative MaOEAs in 20 benchmark problems. Experimental results demonstrate that MaOEA/BnD effectively balances convergence and diversity when handling MaOPs with complex Pareto fronts.

A novel clustering-based evolutionary algorithm with objective space decomposition for multi/many-objective optimization

The framework of decomposing a multi-objective optimization problem (MOP) into some MOPs holds considerable promise. However, its advancement is constrained by numerous elements, including the incorrect segmentation of the subspaces and the challenges in balancing convergence and diversity. To address these issues, an objective space Decomposition and Clustering-based Evolutionary Algorithm (DCEA) is proposed in this paper. Specifically, DCEA employs K-means clustering to create an appropriate mating pool for each individual without the necessity to predetermine the number of clusters. Within each mating pool, the proposed adaptive evolutionary operator is applied to produce offspring for balancing the convergence and diversity. To enhance the accuracy of partitioning, a refined environmental selection approach utilizing supplementary weight vectors is developed. Additionally, by utilizing historical clustering data, a straightforward approach to periodically adjust reference vectors for the allocation of computational resources is proposed. In experiments, both MOPs and many-objective optimization problems (MaOPs) are used to test DCEA. A total of 27 MOPs and 30 MaOPs are involved and 16 state-of-the-art algorithms are employed to compare with DCEA. Comprehensive experiments show that DCEA is an effective algorithm for solving both MOPs and MaOPs.

Merging preferences into the best solution seeking for many-objective optimization problems

Finding out a best solution of the multi-objective optimization problem (MOP) or many-objective optimization problem (MaOP) in the user’s view, is a significant and practical problem. The multi-objective evolutionary algorithms (MOEAs) are applied to solve the MaOPs. Considering many conflicting objectives, they would generate numerous nondominated solutions. Merging the preferences from the user to objectives is an effective way to measure the nondominated solutions and find out the best one (a posterior approach). But the preference relations may be contradictory and highly sparse. In this paper, we propose a process of finding out a best solution with preference relations among attributes (BSPA). In which, we emphasize to test the contradiction and ease the high sparsity of preference relations, then an available preference degree matrix is obtained and merged into the selection of best solution. Based on the proposed process, three methods of NSGA-II-BSPA, P-NSGA-II and P-GA are proposed to solve the best solution of an MOP or MaOP. The experiments are conducted, and the metrics of contradiction probability, feasibility and availability are proposed to measure the BSPA and the vigorous results are obtained. At last, three case studies are adopted to verify the proposed methods’ effectiveness and efficiency. Meanwhile, the hypothesis tests are conducted and the results indicate that the obtained best solutions of proposed methods are satisfiable.

Weight Vector Definition for MOEA/D-Based Algorithms Using Augmented Covering Arrays for Many-Objective Optimization

Many-objective optimization problems are today ever more common. The decomposition-based approach stands out among the evolutionary algorithms used for their solution, with MOEA/D and its variations playing significant roles. MOEA/D variations seek to improve weight vector definition, improve the dynamic adjustment of weight vectors during the evolution process, improve the evolutionary operators, use alternative decomposition methods, and hybridize with other metaheuristics, among others. Although an essential topic for the success of MOEA/D depends on how well the weight vectors are defined when decomposing the problem, not as much research has been performed on this topic as on the others. This paper proposes using a new mathematical object called augmented covering arrays (ACAs) that enable a better sampling of interactions of M objectives using the least number of weight vectors based on an interaction level (strength), defined a priori by the user. The proposed method obtains better results, measured in inverted generational distance, using small to medium populations (up to 850 solutions) of 30 to 100 objectives over DTLZ and WFG problems against the traditional weight vector definition used by MOEA/D-DE and results obtained by NSGA-III. Other MOEA/D variations can include the proposed approach and thus improve their results.

Many-objective optimization based mutual feed scheduling for energy system of integrated energy station

The battery storage system of the integrated energy station contains electric vehicles as the distributed energy storage node of the regional power system can help the regional power system to achieve peak shaving and valley filling. This paper proposes a many-objective optimization based mutual feed model of the energy system of the integrated energy station. The model includes multiple modules of electric vehicle (EV), wind turbine (WT), photovoltaic (PV) and battery energy storage system (BESS). BESS was modeled according to the two behaviors of EVs charging and swapping. The model ensures the role of peak shaving and valley filling for regional power, reduces the loss of battery life in the battery storage system of the integrated energy station, and reduces the operating cost of the battery storage system. In the optimized results, the regional power system load peak-to-valley ratio by 28.72 %, and the daily battery charge/discharge life loss is minimized to 0.00036 %. In addition, for known many-objective optimization problems, this paper also proposes a new meta-heuristic many-objective optimization algorithm the non-dominated sorting hunter prey optimization (NSHPO for short) and compared with other four many-objective optimization algorithms by three indexes (Distribution range, Hypervolume index, and Running time). The simulation results show that compared with other algorithms, the HV metrics of NSHPO are lower by about 1.56 %-7.26 %, Spread metrics are better by about 0.39 %-2.73 %, and the proposed NSHPO algorithm is better in terms of convergence and distributivity.

An adaptive reference vector guided many-objective optimization algorithm based on the pareto front density estimation

The performance of evolutionary algorithms using reference vectors to guide the evolution process mainly depends on the adaptive reference vector update strategy. In order to solve the challenging many-objective optimization problems with irregular Pareto fronts, this paper proposes an adaptive reference vector update strategy based on the Pareto front density estimation, which estimates the true Pareto front by finding sparse regions while ensuring the uniform distribution of reference vectors. In addition, an improved environmental selection strategy using the angle-based neighborhood density estimation has been proposed for estimating the neighborhood density to effectively guide the population evolution. On this basis, this paper proposes an adaptive reference vector guided many-objective optimization algorithm based on Pareto front density estimation (MaOEA-PDE). Experimental results on a large number of benchmark problems show MaOEA-PDE achieves better performance compared with some state-of-the-art algorithms in the literature.

A multi-population evolutionary algorithm based on knowledge transfer for constrained many-objective optimization

ABSTRACT Constrained Many-objective Optimization Problems (CMaOPs) are challenging in handling objectives and constraints simultaneously. Here, a novel Constrained Many-objective Optimization Evolutionary Algorithm (CMaOEA) based on Multi-population, Knowledge transfer and Improved environmental selection called CMaMKI is proposed to handle CMaOPs. The proposed framework evolves a task population to solve the original CMaOP and evolves another population to solve a helper problem derived from the original one. To assist solving the original CMaOP, a knowledge expression and transfer strategy is designed to share useful information in the helper population with the task population. Meanwhile, to balance population convergence, diversity and feasibility, an enhanced environmental selection strategy is devised by combining the ε-constrained technique, θ-dominance and subregional density evaluation. The proposed algorithm is evaluated and contrasted with six state-of-the-art algorithms on a set of benchmark CMaOPs. The experimental results demonstrate the superiority and competitiveness of the proposed method.

Deep and wide search assisted evolutionary algorithm with reference vector guidance for many-objective optimization

Many-objective optimization problems appear in a large many practical cases, but maintaining the convergence and diversity of solutions becomes a big challenge. In response to this, a deep and wide search assisted evolutionary algorithm with reference vector guidance (RVEA-DWC) is proposed. In the algorithm, a novel environmental selection criterion based on substituting Iɛ+ indicator for distance is introduced to enhance the convergence, and once the selected individuals exceed the population size after multiple traversals, those with poor convergence or diversity are randomly eliminated. In addition, to tackle the irregular Pareto front shapes, the invalid reference vectors are deleted regularly, and a certain number of reference vectors with local diversity and global diversity are added, so as to conduct a overall search that combines deep search and wide search to balance exploitation and exploration. An experimental study of 5, 10, 15, 20 objectives is conducted on 60 test instances. The results demonstrate that the proposed algorithm is superior to the other twelve many-objective evolutionary algorithms.

A many-objective evolutionary algorithm under diversity-first selection based framework

Many-objective optimization problems (MaOPs) have attracted wide attention. However, most solving methods prioritize the convergence or take the convergence and diversity into account. This causes the difficulty in balancing convergence and diversity of the population still remains, especially for these problems with irregular Pareto fronts. To this end, we propose a many-objective evolutionary algorithm under diversity-first selection based framework (DSFMO), which first emphasizes the diversity and then considers the convergence. Thus, DSFMO can drive individuals to approach the Pareto front from various directions, and further achieve the balance between convergence and diversity of the population. In this framework, we first construct a candidate pool by selecting these individuals with small crowded degree (i.e., good diversity), in which the crowded degree is evaluated by the global diversity measure based on the objective transferring. Then, the individual in the constructed candidate pool that has the best convergence is selected to enter the next generation, where the convergence is evaluated by the conditional convergence measure that considers the importance of boundary individuals and evolution status. The above two steps are repeatedly executed until the next generation population is formed. In addition, a space projection assisted mating selection is developed to select high-quality parents for variation and further enhance the exploration and exploitation ability of DSFMO. Experiments on 88 test instances have shown that DSFMO can perform best on most test instances. In other words, DSFMO outperforms its seven competitors in balancing convergence and diversity of the population.

Scalable benchmarks and performance measures for dynamic multi-objective optimization

Dynamic multi-objective optimization problems (DMOPs) can be utilized to model certain real-world problems that have a dynamic nature. Algorithms for solving DMOPs can be evaluated and improved by comparing their performance on different benchmarks. However, some existing benchmarks for DMOPs have the limitation of non-uniform weights for decision variables. Additionally, dynamic many-objective optimization problems (DMaOPs) involve more than three objectives, but only a few existing benchmarks can be extended to accommodate DMaOPs. Furthermore, some existing performance measures for DMOPs may not effectively compare the relative performance differences between multiple algorithms or evaluate the search uniformity among different objectives. In this paper, we propose improvements to an existing benchmark for DMOPs by expanding the impact range of decision variables. Moreover, a benchmark framework that can be extended to accommodate DMaOPs is proposed, thus addressing a research gap between the optimization of DMOPs and DMaOPs. Additionally, a set of performance measures for DMOPs are proposed, which can evaluate the relative performance and search uniformity of multi-objective optimization algorithms. By comparing the performance of state-of-the-art and commonly used algorithms on test problems, we can gain a better understanding of the characteristics and strengths and weaknesses of the algorithms and test problems.

Many-objective evolutionary algorithm based on parallel distance for handling irregular Pareto fronts

In recent years, various many-objective evolutionary algorithms (MaOEAs) have been proved to be successful in solving many-objective optimization problems (MaOPs). However, the performance of most MaOEAs is seriously affected when handling MaOPs with irregular Pareto fronts (PFs). In this paper, a new MaOEA variant based on parallel distance (called PDMaOEA) is proposed to solve MaOPs with irregular PFs. Firstly, two new metrics based on parallel distance are designed. The first one termed diversity metric can adapt to irregular PFs. The second one called comprehensive selection metric can consider both diversity and convergence simultaneously. Based on the two metrics, a mating selection method and an environmental selection strategy are proposed. In the mating selection, solutions with good convergence or diversity are chosen to improve the quality of offspring population. In the environmental selection, the selection pressure is significantly enhanced by the two metrics. Experimental study is validated on 19 irregular problems with different shapes of PFs. Performance of the proposed PDMaOEA is compared with six state-of-the-art algorithms. Statistical analysis shows that the proposed approach is competitive in handling MaOPs with irregular PFs.

A Pareto dominance relation based on reference vectors for evolutionary many-objective optimization

Pareto dominance based approach is a classical method for solving multi-objective optimization problems (MOPs). However, as the number of objectives increases, the selection pressure drops sharply. Solutions with good convergence and diversity are hardly obtained. To tackle these issues, this paper proposes a Pareto dominance relation based on reference vectors (called PRV-dominance) for evolutionary many-objective optimization. In PRV-dominance, solutions in the population are divided into several subregions according to a set of uniform reference vectors. To enhance the convergence, a new convergence metric based on the ranking of objective function values is designed to determine the dominance relationship between two solutions. Then, the density in different subregions is considered to maintain the diversity. In order to verify the performance of our approach, WFG and MaF benchmark problems with 3, 5, 8, and 15 objectives are utilized. Experimental results demonstrate that the proposed PRV-dominance outperforms eight existing dominance relations in balancing convergence and diversity. An improved NSGA-II is suggested based on the proposed PRV-dominance, which shows the competitive performance when compared with six other state-of-the-art algorithms in solving many-objective optimization problems (MaOPs). The effectiveness of the proposed PRV-dominance is also verified on two other existing many-objective evolutionary algorithms.

An improved indicator-based two-archive algorithm for many-objective optimization problems

An improved indicator-based two-archive algorithm for many-objective optimization problems

A Kriging-assisted multi-stage evolutionary algorithm for expensive many-objective optimization problems

A Kriging-assisted multi-stage evolutionary algorithm for expensive many-objective optimization problems

A many-objective evolutionary algorithm based on learning assessment and mapping guidance of historical superior information

Abstract Multi-objective optimization algorithms have shown effectiveness on problems with two or three objectives. As the number of objectives increases, the proportion of non-dominated solutions increases rapidly, resulting in insufficient selection pressure. Nevertheless, insufficient selection pressure usually leads to the loss of convergence, too intense selection pressure often results in a lack of diversity. Hence, balancing the convergence and diversity remains a challenging problem in many-objective optimization problems. To remedy this issue, a many-objective evolutionary algorithm based on learning assessment and mapping guidance of historical superior information, referred to here as MaOEA-LAMG, is presented. In the proposed algorithm, an effective learning assessment strategy according to historical superior information based on an elite archive updated by indicator ${I}_{\varepsilon + }$ is proposed, which can estimate the shape of the Pareto front and lay the foundation for subsequent fitness and acute angle-based similarity calculations. From this foundation, to balance the convergence and diversity dynamically, a mapping guidance strategy based on the historical superior information is designed, which contains clustering, associating, and proportional selection. The performance of the proposed algorithm is validated and compared with ten state-of-the-art algorithms on 24 test instances with various Pareto fronts and real-world water resource planning problem. The empirical studies substantiate the efficacy of the results with competitive performance.

Resource-aware multi-criteria vehicle participation for federated learning in Internet of vehicles

Federated learning (FL), as a safe distributed training mode, provides strong support for the edge intelligence of the Internet of Vehicles (IoV) to realize efficient collaborative control and safe data sharing. However, due to the resource limitation and the instability of training environment in the complex IoV, ideal performance of FL cannot be achieved. Since considering the actual resource constraints and federated task requirements, the diversified device selection criteria make the resource-aware vehicle selection problem become a multi-criteria selection problem. To effectively support FL for IoV, the resource-aware multi-criteria vehicle selection problem was described as a many-objective optimization problem, and a resource-aware many-objective vehicle selection model (RA-MaOVSM) is proposed to optimize resource efficiency. The RA-MaOVSM considering heterogeneous resources (like computation resources, communication resources, energy resources and data resources) of on-board devices in IoV, and realizes the joint optimization of learning efficiency, energy cost and global performance. Additionally, a novel probability distribution combination game strategy is applied to many-objective evolutionary algorithm (MaOEA) for improving the model solving performance. Simulation results demonstrate that RA-MaOVSM can effectively optimize the IoV resources and FL model performance, and the designed algorithm exhibits good convergence and distribution, achieving a good balance among multiple device selection criteria.