Many-objective Optimization Problems Research Articles

Moboa: a proposal for multiple objective bean optimization algorithm

The primary objective of multi-objective evolutionary algorithms (MOEAs) is to find a set of evenly distributed nondominated solutions that approximate the Pareto front (PF) of a multi-objective optimization problem (MOP) or a many-objective optimization problem (MaOP). This implies that the approximated solution set obtained by MOEAs should be as close to PF as possible while remaining diverse, adhering to criteria of convergence and diversity. However, existing MOEAs exhibit an imbalance between achieving convergence and maintaining diversity in the objective space. As far as the diversity criterion is concerned, it is still a challenge to achieve an evenly distributed approximation set with different sizes for a problem with a complicated PF shape. Furthermore, Pareto dominance has its own weaknesses as the selection criterion in evolutionary multiobjective optimization. Algorithms based on Pareto criterion (PC) can suffer from problems such as slow convergence to the optimal front and inferior performance on problems with many objectives. To effectively address these challenges, we propose a multi-objective bean optimization algorithm (MOBOA). Given that the selection of parent species, representing global optimal solutions, directly influences the convergence and diversity of the algorithm, MOBOA incorporates a preference order equilibrium parent species selection strategy (POEPSS). By extending the Pareto criterion with the preference order optimization criterion, the algorithm effectively enhances parent species selection pressure across multiple objectives. To balance convergence and diversity, MOBOA proposes a multi-population global search strategy explicitly maintaining an external archive during the search process. Leveraging the inherent multi-population advantages of bean optimization algorithm (BOA), the algorithm facilitates information sharing among the main population, auxiliary populations, and historical archive solution sets. Additionally, a diversity enhancement strategy is employed in the environmental selection stage, introducing the environmental selection strategy of the SPEA2 algorithm to generate a set of evenly distributed nondominated solutions. Experimental results on a series of widely used MOPs and MaOPs demonstrate that the proposed algorithm exhibits higher effectiveness and competitiveness compared to state-of-the-art algorithms.

Two-stage many-objective evolutionary algorithm: enhanced dominance relations and control mechanisms for separated balance

Although the multiobjective evolutionary algorithms (MOEAs) have been proved to bring promising prospects for solving multiobjective optimization problems (MOPs), the performance of the algorithm deteriorates sharply in high-dimensional objective space due to the weak selection pressure and the unregulated balance, which is caused by the increase of objective space dimension. Some current MOEAs with two-stage strategy (TS) strive to address above issues by dividing the evolutionary process into two independent stages, in which convergence and diversity are handled separately within successive generations of different stages. However, TS-MOEAs have some weaknesses, such as sensitivity to stage division, and incomplete separation of convergence and diversity. In this paper, TS/KW-MaOEA is proposed for solving many-objective optimization problems (MaOPs), which keeps TS as the central and equips a perfect control mechanism for separated balance. More specifically, TS/KW-MaOEA can automatically adjust the balance trend and provide appropriate selection pressure for MaOPs according to the Kondratiev wave (KW) search model and the objective space dimension. To verify the effectiveness of the proposed algorithm, a series of experiments are carried out against seven state-of-the-art many-objective optimization algorithms on 15 benchmark problems with up to 30 objectives. Experimental results indicate that the proposed algorithm is highly competitive against peer competitors.

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.

Adaptive normal vector guided evolutionary multi- and many-objective optimization

Most existing multi-objective evolutionary algorithms relying on fixed reference vectors originating from an ideal or a nadir point may fail to perform well on multi- and many-objective optimization problems with various convexity or shapes of Pareto fronts. A possible reason could be the inaccurate measurement of the diversity of solutions or the failure of the fixed reference vectors in guiding the rapidly changing population. To meet this challenge, this work develops an adaptive normal reference vector-based decomposition strategy for guiding the search process, which is able to handle various convexity and shapes of Pareto fronts. Specifically, the normal vector passing through the center of each cluster in a constructed hyperplane is adopted as the reference vector for guiding the search process. Then, a selection strategy is put forward based on the positions of solutions in the current population and the normal vectors for the environmental selection. Based on the adaptive normal vectors, the proposed algorithm can not only rapidly adapt to the changing population but also alleviate the influence of the convexity of Pareto fronts on the measurement of diversity. Experimental results show that the proposed algorithm performs consistently well on various types of multi-/many-objective problems having regular or irregular Pareto fronts. In addition, the proposed algorithm is shown to perform well in the optimization of the polyester fiber esterification process.

Surrogate-Assisted Environmental Selection for Fast Hypervolume-Based Many-Objective Optimization

Hypervolume-based evolutionary algorithms have been widely used to handle many-objective optimization problems. In such algorithms, hypervolume-based environmental selection (HVES), which aims at selecting a subpopulation with the maximal hypervolume (HV) from the current population, plays a crucial role in guiding evolution. However, the computation time of HV increases exponentially with the number of objectives, making the HVES task an expensive optimization problem. In this paper, we propose an efficient surrogate-assisted greedy inclusion algorithm to deal with computationally expensive HVES tasks. It uses a lightweight surrogate model, radial basis function network, to replace the most time-consuming calculations. In addition, an L1-norm distance-based filter is performed as a preselection operator to reduce the search space and avoid some unnecessary calculations. Considering the inevitable approximation errors of surrogate models, we also design an online sampling strategy to enhance the reliability of selected solutions. The proposed algorithm is tested on two types of datasets and compared with six state-of-the-art greedy algorithms. Experimental results show that the proposed algorithm performs excellently on most datasets.

Knowledge transfer based many-objective approach for finding bugs in multi-path loops

AbstractGenerating test cases is essential for discovering software bugs. However, finding bugs in multi-path loops is challenging, especially when bugs can only be exposed after a specific interleaving execution order. Furthermore, the number of paths increases exponentially when the iteration count increases. It results in running out of resources (e.g., allocated search budget) in existing algorithms for test case generation before finding program bugs. To overcome these challenges and improve the effectiveness of existing algorithms in finding bugs among multi-path loops, we propose a path-oriented test case generation framework to explore loop paths of bounded depth. To fulfill this task, we first present a method that groups loop paths based on their latent similarities. Then, we formulate the problem of generating test cases to cover loop paths as a many-objective optimization one. The many-objective optimization problem is designed to optimize one group of loop paths at a time. Finally, a many-objective test case generation framework is used to generate test cases for covering loop paths. In this framework, a knowledge transfer scheme is incorporated to reuse suitable archived test cases at the start of the search process for a similar group of paths. We have implemented the proposed knowledge transfer scheme with generalized differential evolution 3 (KT-GDE3) and evaluated it with benchmarks from the competition on software verification (SV-COMP16 and SV-COMP21). The experimental results show that KT-GDE3 outperforms alternative search-based test case generation algorithms in bug finding capability for multi-path loops. To evaluate the effectiveness of the proposed knowledge transfer scheme, our approach is compared with the variant without this scheme (called KT-GDE3-unarchived) in the benchmarks. The corresponding experimental results verify our assumption that the knowledge transfer scheme can significantly enhance the optimization process of multi-objective algorithms such as generalized differential evolution when finding bugs in multi-path loops.