Irregular Pareto Fronts Research Articles

Many-Objective Grasshopper Optimization Algorithm (MaOGOA): A New Many-Objective Optimization Technique for Solving Engineering Design Problems

In metaheuristic multi-objective optimization, the term effectiveness is used to describe the performance of a metaheuristic algorithm in achieving two main goals—converging its solutions towards the Pareto front and ensuring these solutions are well-spread across the front. Achieving these objectives is particularly challenging in optimization problems with more than three objectives, known as many-objective optimization problems. Multi-objective algorithms often fall short in exerting adequate selection pressure towards the Pareto front in these scenarios and difficult to keep solutions evenly distributed, especially in cases with irregular Pareto fronts. In this study, the focus is on overcoming these challenges by developing an innovative and efficient a novel Many-Objective Grasshopper Optimisation Algorithm (MaOGOA). MaOGOA incorporates reference point, niche preserve and information feedback mechanism (IFM) for superior convergence and diversity. A comprehensive array of quality metrics is utilized to characterize the preferred attributes of Pareto Front approximations, focusing on convergence, uniformity and expansiveness diversity in terms of IGD, HV and RT metrics. It acknowledged that MaOGOA algorithm is efficient for many-objective optimization challenges. These findings confirm the approach effectiveness and competitive performance. The MaOGOA efficiency is thoroughly examined on WFG1-WFG9 benchmark problem with 5, 7 and 9 objectives and five real-world (RWMaOP1- RWMaOP5) problem, contrasting it with MaOSCA, MaOPSO, MOEA/DD, NSGA-III, KnEA, RvEA and GrEA algorithms. The findings demonstrate MaOGOA superior performance against these algorithms.

A switching competitive swarm optimizer for multi-objective optimization with irregular Pareto fronts

In the past two decades, multi-objective particle swarm optimization, as a powerful swarm intelligence paradigm, has been widely used to solve multi-objective optimization problems. By further exploring the competition mechanism among particles in the swarm, some particle update strategies have been proposed, which are effective in improving convergence performance and search speed. However, these strategies encounter difficulties in properly approximating irregular Pareto fronts (PFs), as their geometric structures are complex, leading to get trapped in local optima. To address this issue, a switching competitive swarm optimizer for multi-objective optimization with irregular PFs is proposed. In comparison with existing approaches that utilize reference vectors to guide the search or selection process, the proposed approach introduces a competition-based learning strategy with a novel winner determination mechanism. Within this strategy, independent neighborhoods are constructed for each winner to learn the structure of irregular PFs. To take advantage of the potential knowledge of winners, the switching competitive swarm optimizer is designed to provide more diverse search directions. The experimental results demonstrate the superiority of the proposed algorithm over several state-of-the-art evolutionary algorithms in tackling multi-objective optimization problems with irregular PFs.

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.

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 dual-population-based evolutionary algorithm for multi-objective optimization problems with irregular Pareto fronts

When solving multi-objective optimization problems (MOPs) with irregular Pareto fronts (e.g., disconnected, degenerated, inverted) via evolutionary algorithms, a critical issue is how to obtain a set of well-distributed Pareto optimal solutions. To remedy this issue, we propose a dual-population-based evolutionary algorithm with individual exploitation and weight vector adaptation, named DPEA-IEAW. Specifically, the two populations, termed globPop and locPop, individually evolve by decomposition-based and Pareto dominance-based methods, responsible for global evolution and local evolution, respectively. These two populations collaborate through substantial information exchange, thereby facilitating each other’s evolution. Firstly, the distribution of the two populations is analyzed and an individual exploitation operation is designed for locPop to exploit some promising areas that are undeveloped in globPop. Then, the guide-position is devised for globPop to indicate the optimal point for a subproblem on the Pareto front (PF). By using the guide-position, a strategy for generating uniform weight vectors is proposed to improve the population diversity. Finally, comprehensive experiments on 37 widely used test functions and 2 real-world problems demonstrate that the proposed DPEA-IEAW outperforms comparison algorithms in solving MOPs with various PFs.

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.

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.

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.

A MOEA/D with adaptive weight subspace for regular and irregular multi-objective optimization problems

The performance of the decomposition-based multi-objective optimization algorithm (MOEA/D) is dependent on the consistency of the Pareto front and the distribution of the weight vectors. Uniformly distributed weight vectors have a great advantage in solving Pareto fronts with simplex-like shapes. However, the fronts of real-world problems to be solved are often irregular, and fixed weight vectors are likely to cause deterioration of the solution. To obtain a set of solution sets that are uniformly distributed in the objective space, a MOEA/D-based weight adaptive updating algorithm (called MOEA/D-AWS) is proposed. First, the subproblem evolutionary matrix similarity is used to determine when to adjust the weight vectors. Second, the objective space is divided by creating a subspace of weight vectors to increase population diversity. Third, partial weights are given a second chance to be selected based on the subspace size. Experimental tests are conducted with seven representative algorithms on benchmark functions DTLZ1-7, IDTLZ1-2, MaF1-7, and WFG1-5 (with objective numbers of 3, 5, 8, 10, and 15). The results show that MOEA/D-AWS is more effective for irregular Pareto fronts, especially discontinuous, degenerate, and sharp-tailed Pareto front problems.

A Multi-Objective Evolutionary Algorithm Based on Uniformity and Diversity to Handle Regular and Irregular Pareto Front Shapes

A Multi-Objective Evolutionary Algorithm Based on Uniformity and Diversity to Handle Regular and Irregular Pareto Front Shapes

A solution potential-based adaptation reference vector evolutionary algorithm for many-objective optimization

Decomposition-based multiobjective evolutionary algorithms are widely utilized for solving multiobjective optimization problems (MOPs). These algorithms have a higher degree of diversity due to a uniform distribution of the reference vectors in the objective space. However, these algorithms may struggle to select a suitable candidate set for solving many-objective optimization problems (MaOPs) with irregular Pareto fronts (PFs). To solve this problem, a solution potential-based adaptation reference vector evolutionary algorithm (SPARVEA) is proposed in this paper. In the algorithm, a concept called solution potential is presented to assess whether the direction of convergence of the solution to the ideal solution has potential. The solution potential is obtained with an evaluation function to calculate the potentials of the corresponding solutions. A solution potential-based adaptation strategy is then designed for adapting the reference vector to better guide the direction of convergence of the solution when solving MaOPs with irregular PFs. A modified angle penalty distance (mAPD) is adjusted to improve the convergence rate of the solution set in many-objective optimization problems. The proposed algorithm SPARVEA is compared with state-of-the-art many-objective evolutionary algorithms on four test sets with irregular PFs and four practical problems. Experimental results demonstrate the superior performance of SPARVEA on many-objective optimization problems. This contribution helps to advance the development of efficient algorithms for solving many-objective optimization problems.

Competitive Decomposition-Based Multiobjective Architecture Search for the Dendritic Neural Model.

The dendritic neural model (DNM) is computationally faster than other machine-learning techniques, because its architecture can be implemented by using logic circuits and its calculations can be performed entirely in binary form. To further improve the computational speed, a straightforward approach is to generate a more concise architecture for the DNM. Actually, the architecture search is a large-scale multiobjective optimization problem (LSMOP), where a large number of parameters need to be set with the aim of optimizing accuracy and structural complexity simultaneously. However, the issues of irregular Pareto front, objective discontinuity, and population degeneration strongly limit the performances of conventional multiobjective evolutionary algorithms (MOEAs) on the specific problem. Therefore, a novel competitive decomposition-based MOEA is proposed in this study, which decomposes the original problem into several constrained subproblems, with neighboring subproblems sharing overlapping regions in the objective space. The solutions in the overlapping regions participate in environmental selection for the neighboring subproblems and then propagate the selection pressure throughout the entire population. Experimental results demonstrate that the proposed algorithm can possess a more powerful optimization ability than the state-of-the-art MOEAs. Furthermore, both the DNM itself and its hardware implementation can achieve very competitive classification performances when trained by the proposed algorithm, compared with numerous widely used machine-learning approaches.

An information entropy-driven evolutionary algorithm based on reinforcement learning for many-objective optimization

Many-objective optimization problems (MaOPs) are challenging tasks involving optimizing many conflicting objectives simultaneously. Decomposition-based many-objective evolutionary algorithms have effectively maintained a balance between convergence and diversity in recent years. However, these algorithms face challenges in accurately approximating the complex geometric structure of irregular Pareto fronts (PFs). In this paper, an information entropy-driven evolutionary algorithm based on reinforcement learning (RL-RVEA) for many-objective optimization with irregular Pareto fronts is proposed. The proposed algorithm leverages reinforcement learning to guide the evolution process by interacting with the environment to learn the shape and features of PF, which adaptively adjusts the distribution of reference vectors to cover the PFs structure effectively. Moreover, an information entropy-driven adaptive scalarization approach is designed in this paper to reflect the diversity of nondominated solutions, which facilitates the algorithm to balance multiple competing objectives adaptively and select solutions efficiently while maintaining individual diversity. To verify the effectiveness of the proposed algorithm, the RL-RVEA compared with seven state-of-the-art algorithms on the DTLZ, MaF, and WFG test suites and four real-world MaOPs. The results of the experiments demonstrate that the suggested algorithm provides a novel and practical method for addressing MaOPs with irregular PFs.

A many-objective evolutionary algorithm with metric-based reference vector adjustment

To deal with the inconsistency of the distribution of reference vectors (RVs) and the shape of Pareto front in decomposition-based multiobjective evolutionary algorithms, many literatures proposed to adjust RVs in evolutionary process. However, most of the existing algorithms adjust RVs either in each generation or at a fixed frequency, not considering the evolving information of the population. To tackle this issue, we propose a many-objective evolutionary algorithm with metric-based reference vector adjustment named MBRA. To adjust RVs periodically and conditionally, we use the improvement rate of convergence degree of subproblems computed through d1 distance to reflect the whole convergence of the population. Only when the subproblems are considered convergent on the whole, are RVs allowed to be adjusted: delete the invalid RVs associated with no solutions and add new ones using the farthest solution in the subregion of the most crowded RVs for diversity. Besides, in mating selection and environmental selection, we adopt the sum of objectives and fitness-based sorting based on Iε+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_{\\varepsilon + }$$\\end{document} as the secondary principles to promote convergence, respectively. Validated through extensive experiments, MBRA is effective and competitive on many-objective optimization problems, especially for problems with irregular Pareto fronts.

A Many-Objective Evolutionary Algorithm Based on Non-Linear Dominance

With the increase in the number of objectives, the number of non-dominated solutions will also increase sharply. The sorting method based on the traditional Pareto dominance is not sufficiently distinguishable from the solutions and cannot provide enough selection pressure when the population size is small. In this article, a new non-linear dominance (NLD) method is proposed. The main motivation of this method is from the perspective of storage solutions. The number of solutions is small and the difference between each component is as large as possible, so the part of the first quadrant, the second, and the fourth quadrant near the first quadrant becomes the dominant interval, except for the distance too far also defined as the dominant interval, for which construct a parabolic shape of the non-dominant interval. Based on this relationship, the authors propose a non-linear dominated many-objective evolutionary algorithm (NLDEA), which can solve the irregular Pareto front. Experiments show that NLDEA is competitive with the most advanced methods for various scalable benchmark problems.

An incremental learning evolutionary algorithm for many-objective optimization with irregular Pareto fronts

Multi objective optimization problems (MOPs) with irregular Pareto Fronts (PFs) appear frequently in a large of practical cases. Proximity and diversity are the key indicators for decomposition-based multi-objective evolutionary algorithms (MOEA/D) to solve MOPs, which are decided by the neighborhood structure and a set of predefined direction vectors. In certain MOPs with irregular PFs, the effectiveness of MOEA/D is restricted by the shape of the PF. This limitation arises because MOEA/D generates direction vectors in a uniform manner, which can result in invalid direction vectors due to the irregularity of the PF. When dealing with many-objective optimization problems (MaOP) that have irregular PFs, such as degenerate or disconnected PFs, this decline in effectiveness becomes particularly problematic. To address this limitation, a copula incremental learning (CIL) scheme has been developed to progressively extract implicit knowledge on the appropriate distribution of direction vectors for generating non-uniform direction vectors. Additionally, a niche hierarchical selection (NHS) methodology is employed to construct the neighborhood structure and prevent the generation of duplicate solutions. These enhancements are designed to improve the overall effectiveness of MOEA/D. Efficiency is further ensured through the use of convergence-guided direction (CGD) to approximate irregular PFs. Statistical analysis indicates that the proposed method outperforms other competitive algorithms across most test benchmarks, particularly in its ability to effectively address MaOPs with irregular PFs.

Dynamical decomposition and selection based evolutionary algorithm for many-objective optimization

Decomposition-based many-objective evolutionary algorithms decompose the objective space into multiple subregions, with the help of a set of predefined reference vectors. These vectors serve to guide coevolution between subproblems, and while they show potential for maintaining the diversity of solutions, they have limited exploration capabilities in complex problems and high-dimensional objective space. The main issue is that the predefined reference vectors cannot maintain uniformity of the intersection points between search directions and the irregular Pareto front (PF). To address this problem, this paper proposes a dynamical decomposition and selection strategy (DDS). In DDS, the predefined reference vectors are replaced by solutions themselves and normal-boundary directions (NBI), which guide the population to automatically adapt to the shape of PF. The subregions adapt to divide the objective space to increase diversity. Then, to adjust the relationship between diversity and convergence, a dynamical selection strategy based on the course of evolution is proposed. The process of dynamical decomposition and selection strategy is repeated until the termination condition is met. The proposed algorithm is compared with the state-of-the-art many-objective optimization algorithms on several benchmark problems with 5 to 15 objectives in evolutionary computation. Experimental results show that it outperforms other algorithms in most test instances and is less sensitive to irregular PFs.

A two-stage adaptive reference direction guided evolutionary algorithm with modified dominance relation for many-objective optimization

Traditional dominance-based multi-objective evolutionary algorithms cease to be effective as the number of objectives increases due to the non-dominated sorting mechanism. Accordingly, a novel clustering indicator founded on the penalty-based adaptive rectangular area (APA) between solution and reference direction is proposed to assist in non-dominated levels sorting to deal with this issue. However, directly predefined reference directions with uniform distribution usually cause deteriorated performance in solving multi-objective problems with irregular Pareto fronts. Thus, an adaptive adjustment method guided by the local population is implemented in this paper. At this rate, a two-stage adaptive reference point guided evolutionary algorithm with APA-based dominance relation for many-objective optimization problems (named AREA-APA) is proposed and tested for solving these multi-objective optimization problems (including constrained and unconstrained problems). The proposed algorithm is proven to achieve comparable performance on scalable benchmark problems compared with state-of-the-art algorithms.

Growing neural gas assisted evolutionary many-objective optimization for handling irregular Pareto fronts

Reference vector adjustment approaches pose a considerable potential to strengthen the capability of decomposition-based evolutionary many-objective algorithms in handling many-objective optimization problems (MaOPs) with irregular Pareto-optimal fronts (PFs). However, most existing approaches often rely on the current population’s distribution, which may not well match MaOPs’ PFs. This can easily mislead the reference vector adjustment and impair algorithms’ performance. To address the abovementioned issue, this paper designs a growing neural gas-assisted evolutionary many-objective optimization algorithm, called GNG-EMO. Specifically, the proposal contains an environmental selection without reference vectors to maintain a well-diversified and converged archive over the PFs. Also, the GNG-EMO leverages a growing neural gas (GNG) network to learn Pareto-optimal fronts’ topologies using the solutions in the archive. In addition, to alleviate the issue that the nodes of the GNG network cannot reach the boundaries of Pareto-optimal fronts, we design an expansion mechanism to further diversify the nodes of the GNG network for constructing a set of better reference vectors. To evaluate the performance of the GNG-EMO, we compare it with five state-of-the-art competitors in the context of 48 test cases with irregular PFs. The numerical results demonstrate the GNG-EMO’s superior competitiveness by significantly outperforming all the five competitors on 27 test cases with respect to the indicator Hypervolume.

A Pareto Front grid guided multi-objective evolutionary algorithm

For multi-objective optimization problems with irregular Pareto Fronts, most widely used decomposition methods in MOEA/D (multi-objective evolutionary algorithms based on decomposition) have shown to be lack of the ability to balance the diversity and the convergence to track the true Pareto Fronts during the search. This research investigates the recently proposed grid-based decomposition methods, which reflect the inherent characteristics of the neighborhood structure in the solution to address the issues of diversity and convergence. The performance of the grid-based decomposition method, however, depends on the size of its grid segmentation, and its time complexity increases with the number of grids. In order to improve the computational efficiency, we propose a new concept of Pareto Front grid to guide the search in MOEA. In order to reduce the computing time, a new nadir point estimation strategy based on statistical analysis has been proposed to estimate the whole population. In addition, based on the idea of knee point, a novel grid-based knee point selection method is proposed in the environmental selection of the next generation. Finally, a grid-based decomposition multi-objective evolutionary algorithm with Pareto Front Grid (PFG-MOEA) is proposed. Extensive experimental analysis demonstrates the effectiveness of the proposed PFG-MOEA against state-of-the-art multi-objective evolution algorithms. As the extension of the CDG-MOEA (Constrained Decomposition approach with Grids MOEA) algorithm in the literature, PFG-MOEA can obtain better performance by consuming much less computing time.