Complex Pareto Fronts Research Articles

Research and engineering application of layout optimization for lockbolt structures in railway wagons considering multidimensional failure modes based on MSNSGA-III

The lockbolt structure is essential in railway wagons, and a scientific lockbolt layout can ensure uniform load distribution, thereby preventing failure. However, current engineering lacks layout optimization methods that address multidimensional failure modes. This paper presents a new lockbolt structure layout optimization method based on submodel, parametric models, and a multi-strategy integrated NSGA-III (MSNSGA-III), adhering to the DVS EFB 3435-2 standard. This method simultaneously optimizes the number and spacing of lockbolts to prevent tensile, bearing, shear, and other static failure modes under specified load conditions. The proposed method was applied during the design phase of a container flatcar. Optimization results indicate that, compared to NSGA-III, this method achieves the best IGD and HV values across multiple complex test functions, demonstrating superior performance in solving complex Pareto front optimization problems. Additionally, the optimized lockbolt structure's safety margins increased by a maximum of 59.81%, passing the full vehicle strength test and significantly enhancing resistance to multidimensional failure modes. These results highlight the method's significant practical application value in addressing the optimization of railway wagon lockbolt structures under complex multidimensional failure modes.

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.

Ranking-based multi/many-objective evolutionary optimization with hierarchical evaluation rules and its application in water distribution network

To solve multi/many-objective optimization problems characterized by complex Pareto fronts, we propose a two-phase ranking-based multi-objective evolutionary algorithm with hybrid convergence rules. Firstly, we introduce a novel hybrid convergence criterion by integrating Pareto dominance, localized pruning power (LPP), and distance-based measure. Secondly, we design a two-phase ranking-based selection strategy. Specifically, we employ the acute angle between objective vector and reference vector to divide the current population into multiple sub-populations. In the first-phase ranking, the solutions within each subpopulation are sorted according to the hybrid convergence rule. The best solutions are chosen based on the angle-based diversity criterion into separate fronts. In the second-phase selection, a specific number of solutions with best diversity are further selected to the next generation based upon their front-level assignments. To obtain better diversity of selected solutions, we employ the maximum angle between solutions and the sum of normalized objectives of solutions to assist the selecting process. The experimental results, which encompass benchmark problems involving up to 10 objectives as well as two real-world application instances, validating our proposed algorithm’s competitiveness and effectiveness.

Decomposition-Based Multiobjective Optimization Algorithms With Adaptively Adjusting Weight Vectors and Neighborhoods

The decomposition-based multi-objective optimization algorithm (MOEA/D) is an effective method of solving a multi-objective optimization problem (MOP). The main idea of MOEA/D is that the objectives are weighted through different vectors to form different subproblems, and an optimal solution set is obtained by co-evolution in a certain neighborhood. However, with the increase of objectives, the number of non-dominated solutions increases exponentially, resulting in the deteriorated capability of searching for optimal solutions. In addition, for an optimization problem with the complex Pareto front(PF), the selection pressure of non-dominated solutions is insufficient. To make evolution more efficient, a decomposition-based multi-objective optimization algorithm with adaptively adjusting weight vectors and neighborhoods (MOEA/D-AAWN) is developed in this paper. Firstly, the evolutionary direction of each subproblem is analyzed and the Sparsity function (Spa) is proposed to measure the population density on the PF. By using Spa, a method of generating uniform vectors is presented to improve the diversity of solutions. Besides, a method of adaptively adjusting neighborhoods is given. It adjusts neighborhoods according to the number of iterations and the Spa value of its corresponding subproblem. In this way, the computational resource can be effectively allocated, leading to the improvement in evolutionary efficiency. The proposed algorithm is applied to solve a series of benchmark optimization instances, and the experimental results show that the proposed algorithm outperforms comparison algorithms in runtime, convergence, and diversity.

M-E-AWA: A Novel Task Scheduling Approach Based on Weight Vector Adaptive Updating for Fog Computing

Task offloading and real-time scheduling are hot topics in fog computing. This paper aims to address the challenges of complex modeling and solving multi-objective task scheduling in fog computing environments caused by widely distributed resources and strong load uncertainties. Firstly, a task unloading model based on dynamic priority adjustment is proposed. Secondly, a multi-objective optimization model is constructed for task scheduling based on the task unloading model, which optimizes time delay and energy consumption. The experimental results show that M-E-AWA (MOEA/D with adaptive weight adjustment based on external archives) can effectively handle multi-objective optimization problems with complex Pareto fronts and reduce the response time and energy consumption costs of task scheduling.

A Novel Decomposition-Based Multi-Objective Evolutionary Algorithm with Dual-Population and Adaptive Weight Strategy

Multi-objective evolutionary algorithms mainly include the methods based on the Pareto dominance relationship and the methods based on decomposition. The method based on Pareto dominance relationship will produce a large number of non-dominated individuals with the increase in population size or the number of objectives, resulting in the degradation of algorithm performance. Although the method based on decomposition is not limited by the number of objectives, it does not perform well on the complex Pareto front due to the fixed setting of the weight vector. In this paper, we combined these two different approaches and proposed a Multi-Objective Evolutionary Algorithm based on Decomposition with Dual-Population and Adaptive Weight strategy (MOEA/D-DPAW). The weight vector adaptive adjustment strategy is used to periodically change the weight vector in the evolution process, and the information interaction between the two populations is used to enhance the neighborhood exploration mechanism and to improve the local search ability of the algorithm. The experimental results on 22 standard test problems such as ZDT, UF, and DTLZ show that the algorithm proposed in this paper has a better performance than the mainstream multi-objective evolutionary algorithms in recent years, in solving two-objective and three-objective optimization problems.

Decomposition-based evolutionary algorithm with dual adjustments for many-objective optimization problems

Balancing the convergence and diversity of solutions is a pivotal task for many-objective optimization problems (MaOPs). The decomposition-based evolutionary algorithm has demonstrated great potential in solving MaOPs in the past years. However, its performance degrades when MaOPs have complex Pareto fronts (PFs). Inspired by its pros and cons, this paper proposes a decomposition-based evolutionary algorithm adopting dual adjustments to address MaOPs with irregular PFs. First, an MaOP is divided into a set of subproblems by the distance between weight vectors. Each subproblem selects an appropriate solution from its region, using the specified scalarizing function. Then, the first adjustment updates all the scalarizing functions for each weight vector, where a strategy integrating history information is used to promote the accuracy of the adjustment. Sequentially, the second adjustment updates weight vectors based on the population distribution, which simulates and modifies the value function of reinforcement learning to intensify the rationality of updates. Note that the excitation frequencies of two adjustments are adaptive. Additionally, we design fine-tuning introducing reminding solutions to enhance exploitation. Finally, numerous experiments demonstrate that the proposed algorithm performs better or is equivalent to five state-of-the-art algorithms on 150 test instances and one practical problem . • A novel dual adjustment strategy is proposed, incorporating it into the decomposition-based evolutionary algorithm (DEAdas) to handle the irregular MaOPs. • Adaptive scalarizing functions are presented, and their updates make use of evolutionary information. This adjustment provides an appropriate scalarizing function for each weigh vector. • The adjustment of weight vectors considers the potential of subregions and population distribution, which promotes convergence and diversity. • Experimental results demonstrate that the proposed DEAdas has superiority over peer algorithms.

A new adaptive decomposition-based evolutionary algorithm for multi- and many-objective optimization

In decomposition-based multi-objective evolutionary algorithms (MOEAs), a set of uniformly distributed reference vectors (RVs) is usually adopted to decompose a multi-objective optimization problem (MOP) into several single-objective sub-problems, and the RVs are fixed during evolution. When it comes to multi-objective optimization problems (MOPs) with complex Pareto fronts (PFs), the effectiveness of the multi-objective evolutionary algorithm (MOEA) may degrade. To solve this problem, this article proposes an adaptive decomposition-based evolutionary algorithm (ADEA) for both multi- and many-objective optimization. In ADEA, the candidate solutions themselves are used as RVs, so that the RVs can be automatically adjusted to the shape of the Pareto front (PF). Also, the RVs are successively generated one by one, and once a reference vector (RV) is generated, the corresponding sub-objective space is dynamically decomposed into two sub-spaces. Moreover, a variable metric is proposed and merged with the proposed adaptive decomposition approach to assist the selection operation in evolutionary many-objective optimization (EMO). The effectiveness of ADEA is compared with several state-of-the-art MOEAs on a variety of benchmark MOPs with up to 15 objectives. The empirical results demonstrate that ADEA has competitive performance on most of the MOPs used in this study.

An enhanced decomposition-based multi-objective evolutionary algorithm with a self-organizing collaborative scheme

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multi-objective optimization problem (MOP) into multiple single-objective subproblems using an aggregation function and optimizes them together using a collaborative approach. MOEA/D exhibits extraordinary optimization ability in solving MOPs. However, the algorithm’s performance is hampered by the single evolutionary operator, fixed control parameters, and fixed neighborhood architecture used in primal MOEA/D. To obtain the balance between convergence and diversity, a self-organizing collaborative scheme (SCS) is designed to achieve the best combination of differential evolution (DE) operator, control parameters, and neighborhood size in this paper. The SCS is integrated into MOEA/D to form the MOEA/D-SCS. In MOEA/D-SCS, firstly, the evolutionary process is separated into three stages: early, middle, and late, and different indicators are utilized to screen elite and inferior solutions for the early and late stages. Secondly, different evolutionary strategies are developed for the three stages and the various types of solutions to help boost the algorithm’s efficiency. Then, to enhance convergence and diversity in different evolutionary stages, a neighborhood reconstruction strategy is proposed to assign different neighborhood sizes according to the neighborhood information. Finally, the capability of MOEA/D-SCS was tested using three standard test suites, ZDT, DTLZ, and UF, and a MOP of sewage treatment. The simulation results demonstrate that the proposed MOEA/D-SCS can achieve better optimization results and show stronger optimization ability in most cases compared to state-of-the-art evolutionary algorithms, such as MOEA/D-DE, MOEA/D-STM, MOEA/D-FRRMAB, MOEA/D-DRA, NSGA-II, and dMOPSO, especially in problems with multiple peaks, complex Pareto fronts, and practical engineering optimization problems.

A decomposition-based many-objective evolutionary algorithm with adaptive weight vector strategy

Evolutionary multi-objective optimization methods have become increasingly popular in finding a representative set of Pareto optimal solutions for multi-objective optimization problems (MOPs). As an important multi-objective optimization method, MOEA/D has been successfully applied to resolve various of MOPs. However, there still exist two issues in MOEA/D. First, the distribution of result set is greatly limited in complex Pareto Front (PF) shapes. Second, the effectiveness is relatively poor for solving many-objective optimization problems (MaOPs). Although many researchers attempt to improve MOEA/D, these two issues have not been solved well. This paper proposes a modified MOEA/D method called MOEA/D-VW which involves three optimization strategies. First, an adaptive weight vector adjustment strategy is adopted to fine-tune the subproblems, thus changing the evolution direction of the subpopulation. Second, a weight vector initialization strategy with preference parameters is added into MOEA/D-VW. Third, a modified crossover operator is leveraged to generate new population with good diversity and overcome the problem of high computational cost. Experiments are carried out on the set of 23 continuous optimization problems. The experimental results show that the performance and the distribution of the Pareto Set (PS) of MOEA/D-VW are effectively improved. Meanwhile, the experimental results demonstrate that the modified crossover operator can effectively reduce the computational cost of MOEA/D-VW.

Multi-objective evolutionary optimization based on online perceiving Pareto front characteristics

When solving multi-objective optimization problems with complex Pareto fronts characteristics, previous work generally ignores the information related to Pareto fronts provided by the population during the evolution, which is detrimental to efficiently tackle them. In order to taking full advantage of information associated with a population evolution, a multi-objective evolutionary optimization method based on online perceiving Pareto front characteristics is proposed in this study. To this end, the information associated with the Pareto front of an optimization problem is first extracted from the population. Following that, the characteristics of the Pareto front in concavity/convexity and continuity are perceived online. For the purpose of each sub-front containing only one characteristic, a Pareto front is divided based on the concavity/convexity and continuity. According to the characteristic of each sub-front, different reference points are selected to refine the distribution of reference vectors. Finally, a multi-objective evolutionary algorithm is designed targeting the characteristics of the Pareto front. The performance of the proposed method is evaluated by comparing it with 8 state-of-the-art optimizers on 31 test problems. Further, the experimental results demonstrate that the proposed method is competitive in handling multi-objective optimization problems with irregular Pareto fronts.

On the Effect of the Cooperation of Indicator-Based Multiobjective Evolutionary Algorithms

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries.

Two-Population Coevolutionary Algorithm with Dynamic Learning Strategy for Many-Objective Optimization

Due to the complexity of many-objective optimization problems, the existing many-objective optimization algorithms cannot solve all the problems well, especially those with complex Pareto front. In order to solve the shortcomings of existing algorithms, this paper proposes a coevolutionary algorithm based on dynamic learning strategy. Evolution is realized mainly through the use of Pareto criterion and non-Pareto criterion, respectively, for two populations, and information exchange between two populations is used to better explore the whole objective space. The dynamic learning strategy acts on the non-Pareto evolutionary to improve the convergence and diversity. Besides, a dynamic convergence factor is proposed, which can be changed according to the evolutionary state of the two populations. Through these effective heuristic strategies, the proposed algorithm can maintain the convergence and diversity of the final solution set. The proposed algorithm is compared with five state-of-the-art algorithms and two weight-sum based algorithms on a many-objective test suite, and the results are measured by inverted generational distance and hypervolume performance indicators. The experimental results show that, compared with the other five state-of-the-art algorithms, the proposed algorithm achieved the optimal performance in 47 of the 90 cases evaluated by the two indicators. When the proposed algorithm is compared with the weight-sum based algorithms, 83 out of 90 examples achieve the optimal performance.

Adaptively weighted decomposition based multi-objective evolutionary algorithm

Multi-objective evolutionary algorithm based on Decomposition (MOEA/D) decomposes a multi-objective problem into a number of scalar optimization problems using uniformly distributed weight vectors. However, uniformly distributed weight vectors do not guarantee uniformity of solutions on approximated Pareto-Front. This study proposes an adaptive strategy to modify these scalarizing weights after regular intervals by assessing the crowdedness of solutions using crowding distance operator. Experiments carried out over several benchmark problems with complex Pareto-Fronts show that such a strategy helps in improving the convergence and diversity of solutions on approximated Pareto-Front. Proposed algorithm also shows better performance when compared with other state-of-the-art multi-objective algorithms over most of the benchmark problems.

Self-adaptive weight vector adjustment strategy for decomposition-based multi-objective differential evolution algorithm

In multi-objective and many-objective optimization, weight vectors are particularly crucial to the performance of decomposition-based optimization algorithms. The uniform weight vectors are not suitable for complex Pareto fronts (PFs), so it is necessary to improve the distribution of weight vectors. Besides, the balance between convergence and diversity is a difficult issue as well in multi-objective optimization, and it becomes increasingly important with the augment of the number of objectives. To address these issues, a self-adaptive weight vector adjustment strategy for decomposition-based multi-objective differential evolution algorithm (AWDMODE) is proposed. In order to ensure that the guidance of weight vectors becomes accurate and effective, the adaptive adjustment strategy is introduced. This strategy distinguishes the shapes and adjusts weight vectors dynamically, which can ensure that the guidance of weight vectors becomes accurate and effective. In addition, a self-learning strategy is adopted to produce more non-dominated solutions and balance the convergence and diversity. The experimental results indicate that AWDMODE outperforms the compared algorithms on WFG suites test instances, and shows a great potential when handling the problems whose PFs are scaled with different ranges in each objective.

Approximating Complex Pareto Fronts With Predefined Normal-Boundary Intersection Directions

Decomposition-based evolutionary algorithms using predefined reference points have shown good performance in many-objective optimization. Unfortunately, almost all experimental studies have focused on problems having regular Pareto fronts (PFs). Recently, it has been shown that the performance of such algorithms is deteriorated when facing irregular PFs, such as degenerate, discontinuous, inverted, strongly convex, and/or strongly concave fronts. The main issue is that the predefined reference points may not all intersect with the PF. Therefore, many researchers have proposed to update the reference points with the aim of adapting them to the discovered Pareto shape. Unfortunately, the adaptive update does not really solve the issue for two main reasons. On the one hand, there is a considerable difficulty to set the time and the frequency of updates. On the other hand, it is not easy to define how to update the search directions for an unknown PF shape. This article proposes to approximate irregular PFs using a set of predefined normal-boundary intersection (NBI) directions. The main motivation behind this article is that when using a set of well-distributed NBI directions, all these directions intersect with the PF regardless of its shape, except for the case of discontinuous and/or degenerate fronts. To handle the latter cases, a simple interaction mechanism between the decision maker (DM) and the algorithm is used. In fact, the DM is asked if the number of NBI directions needs to be increased in some stages of the evolutionary process. If so, the resolution of the NBI directions that intersect the PF is increased to properly cover discontinuous and/or degenerate PFs. Our experimental results on benchmark problems with regular and irregular PFs, having up to fifteen objectives, show the merits of our algorithm when compared to eight of the most representative state-of-the-art algorithms.

A modified PBI approach for multi-objective optimization with complex Pareto fronts

Abstract The penalty-based boundary intersection (PBI) approach is widely used in the decomposition-based multi-objective evolutionary algorithm (MOEA/D). Generally, a uniform distribution of weight vectors in PBI approach will lead to a set of evenly distributed solutions on the Pareto-optimal front (POF), but this approach cannot work well in practice when the target multi-objective optimization problem (MOP) has a complex POF. For example, the POF may have disconnected regions and a long tail and a sharp peak and a degenerate geometry, which significantly degrades the performance of the original MOEA/D. This paper proposes a modified PBI (MPBI) approach and a strategy of adjusting reference points (ARP) to handle these MOPs with complex fronts. A two-stage strategy is adopted in the proposed algorithm. The first stage is to determine a hyperplane based on the modified PBI approach, so that the projection points derived from the solutions obtained in second stage to this hyperplane are all in the first quadrant. Exploring those regions where the solution exists is also a key task in this stage. The second stage is to adjust the reference points periodically so that the reference points can be redistributed adaptively to improve the distribution of solutions. The framework of the proposed algorithm is based on θ -DEA and named NSGA-MPBI. Some widely used test instances and three many-objective MOPs with complex POFs are employed in the experiments. The experimental results indicate that NSGA-MPBI outperforms the state-of-the-art algorithms.

MOEA/HD: A Multiobjective Evolutionary Algorithm Based on Hierarchical Decomposition.

Recently, numerous multiobjective evolutionary algorithms (MOEAs) have been proposed to solve the multiobjective optimization problems (MOPs). One of the most widely studied MOEAs is that based on decomposition (MOEA/D), which decomposes an MOP into a series of scalar optimization subproblems, via a set of uniformly distributed weight vectors. MOEA/D shows excellent performance on most mild MOPs, but may face difficulties on ill MOPs, with complex Pareto fronts, which are pointed, long tailed, disconnected, or degenerate. That is because the weight vectors used in decomposition are all preset and invariant. To overcome it, a new MOEA based on hierarchical decomposition (MOEA/HD) is proposed in this paper. In MOEA/HD, subproblems are layered into different hierarchies, and the search directions of lower-hierarchy subproblems are adaptively adjusted, according to the higher-hierarchy search results. In the experiments, MOEA/HD is compared with four state-of-the-art MOEAs, in terms of two widely used performance metrics. According to the empirical results, MOEA/HD shows promising performance on all the test problems.

Approximating the irregularly shaped Pareto front of multi-objective reservoir flood control operation problem

Decomposition based multi-objective evolutionary algorithm (MOEA/D) has been proved to be effective on multi-objective optimization problems. However, it fails to achieve satisfactory coverage and uniformity on problems with irregularly shaped Pareto fronts, like the reservoir flood control operation (RFCO) problem. To enhance the performance of MOEA/D on the real-world RFCO problem, a Pareto front relevant (PFR) decomposition method is developed in this paper. Different front the decomposition method in the original MOEA/D which is based on a unique reference point (i.e. the estimated ideal point), the PFR decomposition method uses a set of reference points which are uniformly sampled from the fitting model of the obtained Pareto front. As a result, the PFR decomposition method can provide more flexible adaptation to the Pareto front shapes of the target problems. Experimental studies on benchmark problems and typical RFCO problems at Ankang reservoir have illustrated that the proposed PFR decomposition method significantly improves the adaptivity of MOEA/D to the complex Pareto front shape of the RFCO problem and performs better both in terms of coverage and uniformity.

An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been recognized as a promising method for solving multi-objective optimization problems (MOPs), receiving a lot of attention from researchers in recent years. However, its performance in handling MOPs with complicated Pareto fronts (PFs) is still limited, especially for real-world applications whose PFs are often complex featuring, e.g., a long tail or a sharp peak. To deal with this problem, an improved MOEA/D (named iMOEA/D) that mainly focuses on bi-objective optimization problems (BOPs) is therefore proposed in this paper. To demonstrate the capabilities of iMOEA/D, it is applied to design optimization problems of truss structures. In iMOEA/D, the set of the weight vectors defined in MOEA/D is numbered and divided into two subsets: one set with odd-weight vectors and the other with even-weight vectors. Then, a two-phase search strategy based on the MOEA/D framework is proposed to optimize their corresponding populations. Furthermore, in order to enhance the total performance of iMOEA/D, some recent developments for MOEA/D, including an adaptive replacement strategy and a stopping criterion, are also incorporated. The reliability, efficiency and applicability of iMOEA/D are investigated through seven existing benchmark test functions with complex PFs and three optimal design problems of truss structures. The obtained results reveal that iMOEA/D generally outperforms MOEA/D and NSGA-II in both benchmark test functions and real-world applications.