Constrained Multi-objective Optimization Problem Research Articles

Spectral projected subgradient method with a 1-memory momentum term for constrained multiobjective optimization problem

Spectral projected subgradient method with a 1-memory momentum term for constrained multiobjective optimization problem

Constrained Multiobjective Optimization with Escape and Expansion Forces

Constraints may scatter the Pareto optimal solutions of a constrained multiobjective optimization problem (CMOP) into multiple feasible regions. To avoid getting trapped in local optimal feasible regions or a part of the global optimal feasible regions, a constrained multiobjective evolutionary algorithm (CMOEA) should consider both the escape force and the expansion force carefully during the search process. However, most CMOEAs fail to provide these two forces effectively. As a remedy for this limitation, this article proposes a method called TPEA. TPEA maintains three populations, termed Pop1, Pop2, and Pop3. Pop1 is a regular population, updated with a constrained NSGA-II variant. Pop2 and Pop3 are two auxiliary populations, containing the innermost and outermost nondominated infeasible solutions, respectively. The analysis reveals that these two types of nondominated infeasible solutions can contribute to the generation of escape and expansion forces, respectively. Due to these two forces, TPEA is likely to identify more global optimal feasible regions, which is crucial for constrained multiobjective optimization. Also, a mating selection strategy is developed in TPEA to coordinate the interaction among these three populations. Extensive experiments on 58 benchmark CMOPs and 35 real-world ones demonstrate that TPEA is significantly superior or comparable to six state-of-the-art CMOEAs on most test instances.

A Two-Phase Kriging-Assisted Evolutionary Algorithm for Expensive Constrained Multiobjective Optimization Problems

A Two-Phase Kriging-Assisted Evolutionary Algorithm for Expensive Constrained Multiobjective Optimization Problems

Constrained Probabilistic Pareto Dominance for Expensive Constrained Multiobjective Optimization Problems

Constrained Probabilistic Pareto Dominance for Expensive Constrained Multiobjective Optimization Problems

A constrained multi-objective optimization algorithm based on coordinated strategy of archive and weight vectors

When dealing with Constrained Multi-objective Optimization Problems (CMOPs) and struggling to enhance feasibility, convergence and diversity, the researchers of Constrained Multi-objective Optimization Evolutionary Algorithms (CMOEAs) gravitate toward feasibility or take precedence to preserve well-converged solutions ignoring diversity in the past. To compensate for the defects, the paper proposes CMOEA-MSWA to guide the search of infeasible regions by coordinated strategy of archive and weight vectors. Firstly, the archive carrying the information of population diversity updates the weight vectors. Secondly, the updated weight vectors perpetuate the diversity information to the search of infeasible regions. The circular effects between strategies promote the detection of infeasible solutions with good objectives and exhibit competitive performance in terms of spread and evenness. To testify the versatility the CMOEA-MSWA in enhancing diversity, the comprehensive performance is evaluated firstly and the diversity analysis of the CMOEA-MSWA is carried out on four benchmark suites with 34 test instances, where the number of objectives for some of test problems is scaled from three to five. In comparison with five state-of-the-art CMOEAs, the proposed algorithm yields highly competitive performance in diversity on different types of CMOPs. In addition, the effectiveness of collaboration between archive and weight vectors on handling infeasible solutions is also verified.

Characterization of geological uncertainties from limited boreholes using copula-based coupled Markov chains for underground construction

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes. Two Markov chains are used to describe the soil transitions along different directions, and the transition probability matrices (TPMs) of the Markov chains are analytically expressed by copulas. This copula expression is efficient since it can represent a large TPM by a few unknown parameters. Due to the analytical expression of the TPMs, the likelihood function of the Markov chain model is given in an explicit form. The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints. Unlike the method which determines the TPMs by counting the number of transitions between soil types, the proposed method is more statistically sound. Moreover, a random path sampling method is presented to avoid the directional effect problem in simulations. The soil type at a location is inferred from its nearest known neighbors along the cardinal directions. A general form of the conditional probability, based on Pickard's theorem and Bayes rule, is presented for the soil type generation. The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan, China. It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation.

A niche-based evolutionary algorithm with dual cooperative archive for solving constrained multi-objective optimization problems

Constrained multi-objective optimization problems (CMOPs) are commonly encountered in engineering practice. The key to effectively solving these problems lies in achieving a timely balance between convergence, diversity, and feasibility during iterations. Furthermore, the appropriate utilization of infeasible solutions is crucial for identifying potential feasible regions. In order to accomplish this comprehensive objective, we propose a novel dual-stage constrained multi-objective evolutionary algorithm (CMOEA) called NACMOEA in this paper. It can be characterized by the following features: 1) Introducing a novel niche-based individual selection and infeasible solution utilization strategy to enhance convergence, diversity, and feasibility. 2) Presenting a cooperative search strategy assisted by dual archives to approximate the constrained Pareto front (CPF) from both feasible and infeasible perspectives, thereby improving the efficiency of obtaining the complete CPF. 3) Designing a new stage switch method based on non-dominant coverage rate to ensure proper completion of search stage switching. Extensive experiments demonstrate that NACMOEA exhibits competitive comprehensive performance when compared with other advanced CMOEAs.

A constrained multi-objective evolutionary algorithm with clustering based weight vector adaptation

In constrained multi-objective optimization, balancing convergence, diversity, and feasibility of solutions in the population are challenging. Most existing approaches focus on convergence and feasibility but fail to enforce the diversity and uniformity of solutions in the approximated Pareto Front (PF). In decomposition-based approaches, the diversity and uniformity of the obtained solution set depend on the set of weight vectors employed, which are generally fixed and initialized uniformly. However, Constrained Multi-objective Optimization Problems (CMOPs) are characterized by discontinuous Pareto-optimal fronts due to the presence of constraints. Therefore, to maintain better diversity and uniformity in the population, the weight vector set employed in the framework should consider the characteristics of PF, which is not known in advance. However, during the evolution, the characteristics of PF can be estimated and the associated weight vectors can be appropriately distributed. Motivated by this issue, we propose a novel Constrained Multi-Objective Evolutionary Algorithm with a Clustering-based Weight vector Adaptation strategy called CMOEA-CWA. The proposed CMOEA-CWA uses the basic framework of CMOEA-DPMS. After exploring the search space, CMOEA-CWA reinitializes the weight vectors if there exist any discontinuities in the approximated pareto front (PF). In CMOEA-CWA, the PF is approximated using the feasible solutions stored in an archive and novel parameter-free clustering techniques. During the clustering process, if there exist any discontinuities in the approximated PF, one population will exploit the feasible regions with a re-initialized set of weight vectors while the other population will explore the feasible regions with uniformly distributed weight vectors. Empirical results obtained on four popular and widely used benchmarks when compared with nine latest state-of-the-art algorithms (ARMOEA, CCMO, TiGE-2, ToP, MOEA/D-DAE, PPS, CTAEA, MOEA/D-2WA and CMOEA-DPMS) suggest the superiority of the proposed algorithm, especially on 47 complicated constrained multi-objective problems. The proposed algorithm performs statistically better than or comparable (in terms of HV) to ARMOEA, CCMO, TiGE-2, ToP, MOEA/D-DAE, PPS, CTAEA, MOEA/D-2WA and CMOEA-DPMS in 91.48 %, 78.72 %, 100 %, 70.21 %, 85.11 %, 78.72 %, 91.49 % and 82.97 % of test instances, respectively. The performance of the proposed algorithm is also demonstrated on a real-world Economic Environmental Dispatch (EED) problem.

A dual-population algorithm based on self-adaptive epsilon method for constrained multi-objective optimization

Balancing multiple objectives and various constraints is crucial for effectively solving constrained multi-objective optimization problems (CMOPs). Excessive focus on either convergence or feasibility may not result in favorable outcomes of the algorithm. To confront this challenge, this paper proposes a cooperative evolutionary algorithm named SaE-CMO, which aims to achieve a harmonious balance between convergence and feasibility by extracting valuable information from both feasible and infeasible regions. To achieve this, SaE-CMO employs a dual-population approach to enhance search progress, consisting of a main population, Population1, and an auxiliary population, Population2. These two populations complement each other to achieve optimal results. A newly proposed self-adaptive epsilon method is employed in both Population1 and Population2, using different comparison criteria to select next population from mating pools, respectively. Population2 can retain some solutions that are well-constrained but poorly converged, thereby preserving information about both the constrained and the unconstrained Pareto front. This property enables Population2 to assist Population1 in maintaining diversity in certain complex CMOPs. To verify the effectiveness of SaE-CMO, we conduct experiments on three benchmark test instances and four real-world CMOPs with some related state-of-the-art constrained multi-objective optimization algorithms, experimental results prove that the proposed algorithm outperforms the compared algorithms.

Bi-directional search based on constraint relaxation for constrained multi-objective optimization problems with large infeasible regions

How to balance the satisfaction of constraints and the optimization of objective functions is one of the key issues to solve constrained multi-objective optimization problems (CMOPs), especially when the constraints are complex. Although many algorithms have been designed to handle this, most of them are still unable to effectively handle CMOPs with complex constraints. Based on the above issue, this paper proposes a framework for bi-directional search, which evolves two populations (P1 and P2). P1 aims to search the constrained Pareto front (CPF) from the infeasible side of the objective space, and P2 from the feasible side, aiming to achieve a more efficient and comprehensive bi-directional search for the CPF. To ensure the diversity, we adopt a preferred weight vector selection strategy to choose potential mating parents, which improves the search capability for the marginal CPF. Furthermore, to coordinate the interaction between the two populations, we propose an environmental selection strategy to select the offspring generated by P1 and P2 under the same weight vector with the better fitness to update the populations respectively, and the fitness is evaluated based on the different constraint relaxations of the two populations, to update them respectively. Extensive experiments indicate that our proposed algorithm has significantly better results or was at least competitive when compared to eight state-of-the-art methods.

A dynamic constrained multiobjective evolutionary algorithm based on decision variable classification

In dynamic constrained multiobjective optimization problems (DCMOPs), dynamics may arise from time-varying objective functions or/and constraints. To solve these problems, maintaining a good balance among feasibility, convergence and diversity of a population under dynamic environments is a critical challenge. Although appropriately utilizing the characteristic of decision variables can promote algorithms to better track the Pareto optima under dynamic environments, their sensitivity to constraints is neglected. Therefore, a dynamic constrained multiobjective evolutionary algorithm based on decision variable classification (DC-MOEA-DVC) is proposed. Under each environment, decision variables are classified into four types in terms of their influence on convergence, distribution, and constraint violation. Based on them, a new offspring generation method is developed, decision variables with different characteristics are rationally combined to generate offspring, with the purpose of accelerating the convergence of the population. Once an environmental change appears, a hybrid strategy consisting of four change response techniques is introduced for the corresponding types of decision variables, producing a new initial population. The experimental results show that DC-MOEA-DVC is superior to the other five state-of-the-art algorithms.

Enhanced auxiliary population search for diversity improvement of constrained multiobjective coevolutionary optimization

The challenge of solving constrained multiobjective optimization problems (CMOPs) is to obtain solution sets that satisfy all constraints and have global optimal convergence and diversity. However, maintaining the distribution of solutions in small and discrete feasible regions while balancing convergence and feasibility has become increasingly challenging when optimizing many complex CMOPs. To address this issue, this paper proposes an enhanced auxiliary population search, EAPS, based on coevolutionary optimization to improve the distribution of the main population in solving complex CMOPs. Specifically, the auxiliary population adopts a new aggregation method under the decomposition framework to evolve independently without considering any constraints and provides favorable diversity information uni-directionally to the main population. The proposed algorithm is compared with five state-of-the-art constrained multiobjective algorithms on four well-known constraint test suites. The experimental results demonstrate that the proposed are competitive or comparable with the comparison algorithms.

A novel competitive constrained dual-archive dual-stage evolutionary algorithm for constrained multiobjective optimization

In constrained multi-objective optimization problems (CMOPs), the key problem is how to balance convergence and diversity simultaneously and find as many feasible solutions as possible in the reduced search space. To address this issue, a novel competitive constrained dual-archive dual-stage evolutionary algorithm, named c-DADSEA, is proposed with a convergence-driven archive (CA) and a diversity-driven archive (DA). To improve the algorithm’s convergence, instead of simply excluding infeasible solutions, an adaptive penalty function is novelly proposed to maintain competitively infeasible solutions in CA. To overcome the difficulty in finding the true Pareto front for the dual-population framework in some problems, a dual-stage framework is introduced to make DA search in different directions at different stages (i.e., Forward and Backward), enhancing the diversity of the algorithm. DA evolves without considering the constraints to explore all regions as much as possible in the Forward stage. While in the Backward stage, DA degenerates with increasing constraints to search the regions around the true Pareto front by using ϵ-constrained method. Moreover, to balance convergence and diversity, different benchmark rankings are designed for the individuals in different archives. The experiments on three benchmark CMOPs test suites demonstrate that c-DADSEA is better than seven other state-of-the-art constrained evolutionary algorithms, especially for the CMOPs with a constrained Pareto front away from the unconstrained Pareto front.

A multi-preference-based constrained multi-objective optimization algorithm

When tackling constrained multi-objective optimization problem, evolutionary algorithms grapple with the simultaneous need to optimize the conflict objectives and satisfy constraints. The preference for the algorithm in processing objectives and constraints directly affects the feasibility, diversity and convergence of population. At the same time, the decision regarding algorithmic preference should be based on the characteristics of the current evolution. Therefore, preference decision of algorithm is a intractable challenge throughout the optimization. To address this issue, a multi-preference-based constrained multi-objective optimization algorithm is proposed in this paper, operating under the aegis of three evolutionary models. The preferences are determined by the analysis of the evolution states in concert with the actual characteristic of the population, and are implemented through the reasonable scheduling evolutionary models. When the preference changes, the algorithm identifies the useful knowledges carried by the previous populations and transfers them to the current populations. In addition, a shift-based update strategy and a new co-evolution strategy are designed for different models in the proposed algorithm, respectively. Compared to five state-of-the-art multi-objective algorithms on two benchmark suits and two real-world applications, the proposed algorithm performs better than its peers, especially in diversity difficulty and convergence difficulty multi-objective optimization problems.

A novel multi-objective evolutionary algorithm with a two-fold constraint-handling mechanism for multiple UAV path planning

Most multiple unmanned aerial vehicle (UAV) path planning problems are often treated as constrained single-objective optimization problems. How to consider them as constrained multi-objective optimization problems (CMOPs) have seldom been explored in this field. To fill this gap, this paper firstly constructs multiple UAV path planning problem as a CMOP with two objectives and five constraints. Then a novel multi-objective evolutionary algorithm with a two-fold constraint-handling mechanism is proposed for multiple UAV path planning. To cope with constraints effectively, a constraint-handling technique based on a progressive weight vector strategy is proposed. Besides, a constraint repair technique that considers the flying environment is designed to further guide the algorithm to find feasible promising regions. Eight multiple UAV path planning test instances with different solving difficulties are constructed. Subsequently, they are used to validate the performance of the proposed algorithm. The experimental results demonstrate that the proposed algorithm is superior over four compared algorithms in terms of obtaining a set of better-distributed Pareto optimal solutions.

An Instance Space Analysis of Constrained Multiobjective Optimization Problems

Constrained multi-objective optimization problems (CMOPs) are generally more challenging than unconstrained problems. This in part can be attributed to the infeasible region generated by the constraint functions, the interaction between constraints and objectives, or both. In this paper, we explore the relationship between the performance of constrained multi-objective evolutionary algorithms (CMOEAs) and the instance characteristics of CMOP using Instance Space Analysis (ISA). To do this, we extend recent work on Landscape Analysis features for characterising CMOPs. Specifically, we introduce new features to describe the multi-objective-violation landscape, formed by the interaction between constraint violation and multi-objective fitness. Detailed evaluation of the algorithm footprints, spanning eight CMOP benchmark suites and fifteen CMOEAs, demonstrates that ISA effectively captures the strength and weakness of the CMOEAs. We conclude that two characteristics, the isolation of non-dominate set and the correlation between constraints and objectives evolvability, have the greatest impact on algorithm performance. However, the current benchmarks problems lack of diversity to represent the real-world problems and to fully reveal the efficacy of CMOEAs evaluated.

A Competitive and Cooperative Swarm Optimizer for Constrained Multiobjective Optimization Problems

Solving multi-objective optimization problems through meta-heuristic methods gets considerable attention. Based on the classical variation operators, several enhanced operators, as well as multi-objective optimization evolutionary algorithms, have been developed. Among these operators, the competitive swarm optimizer exhibits promising performance. However, it encounters difficulties when tackling constrained multi-objective optimization problems with large objective spaces or complex infeasible regions. In this paper, a competitive and cooperative swarm optimizer is proposed, which contains two particle update strategies: i) the competitive swarm optimizer provides faster convergence speed to accelerate the approximation of the Pareto front; and ii) the cooperative swarm optimizer suggests a mutual-learning strategy to enhance the ability to jump out of local feasible regions or local optima. We also present a new algorithm for constrained multi-objective optimization problems. The results on four benchmark suites with 47 instances demonstrate the superiority of our approach compared with other state-of-the-art methods. Additionally, its effectiveness on large scale constrained multi-objective optimization problems has also been verified.

Multiobjective-Based Constraint-Handling Technique for Evolutionary Constrained Multiobjective Optimization: A New Perspective

Multiobjective-based constraint-handling techniques are popular in evolutionary constrained single-objective optimization. However, most of these techniques run into troubles when dealing with constrained multiobjective optimization problems (CMOPs). That is, they have difficulty optimizing too many objective functions, are ineffective in maintaining population diversity, or are challenged in establishing appropriate additional objective functions. As a remedy to these limitations, we propose a novel technique called NRC for handling CMOPs. The novelty of NRC lies in its three sorting procedures: nondominated sorting, reversed nondominated sorting, and constrained crowding distance sorting, which are performed in sequence to provide driving forces toward the Pareto front (PF) of a transformed unconstrained multiobjective optimization problem (treating the overall constraint violation as an additional objective function), the boundary front, and the constrained PF, respectively. With the combination of these three different forces, NRC can conveniently approach the desired PF from diverse search directions. The effectiveness of NRC is experimentally verified. Also, we incorporate NRC into a two-archive mechanism and develop a novel constrained multiobjective evolutionary algorithm, called NRC2. Comprehensive experiments on 49 benchmark CMOPs and 21 real-world ones demonstrate that NRC2 is significantly superior or comparable to six state-of-the-art constrained evolutionary multiobjective optimizers on most test instances.

A Multistage Algorithm for Solving Multiobjective Optimization Problems With Multiconstraints

There are usually multiple constraints in constrained multi-objective optimization. Those constraints reduce the feasible area of the constrained multi-objective optimization problems (CMOPs) and make it difficult for current multi-objective optimization algorithms (CMOEAs) to obtain satisfactory feasible solutions. In order to solve this problem, this paper studies the relationship between constraints, then obtains the priority between constraints according to the relationship between the Pareto Front (PF) of the single constraint and their common PF. Meanwhile, this paper proposes a multi-stage CMOEA and applies this priority, which can save computing resources while helping the algorithm converge. The proposed algorithm completely abandons the feasibility in the early stage to better explore the objective space, and obtains the priority of constraints according to the relationship; Then the algorithm evaluates a single constraint in the medium stage to further explore the objective space according to this priority, and abandons the evaluation of some less-important constraints according to the relationship to save the evaluation times; At the end stage of the algorithm, the feasibility will be fully considered to improve the quality of the solutions obtained in the first two stages, and finally get the solutions with good convergence, feasibility, and diversity. The results on five CMOP suites and three real-world CMOPs show that the algorithm proposed in this paper can have strong competitiveness in existing constrained multi-objective optimization.

A dual-population evolutionary algorithm based on dynamic constraint processing and resources allocation for constrained multi-objective optimization problems

Constrained multi-objective optimization problems (CMOPs) contain the satisfaction of various constraints and optimization of multiple objectives simultaneously, thus they are extremely challenging. Although many constrained multi-objective evolutionary algorithms (CMOEAs) have been proposed, they ignore the information of each constraint, which might help utilize more various infeasible solutions to improve the search ability of the population. Therefore, this paper proposes a new dual-population CMOEA to solve CMOPs, in which a dynamic constraint processing mechanism and a dynamic resource allocating scheme are designed. To be specific, the proposed algorithm evolves two populations, which adopt different mechanisms to handle constraints respectively. The main population directly optimizes all constraints to find the feasible Pareto optimal solutions, which can improve the feasibility. The auxiliary population adopts a dynamic constraint processing mechanism, which gradually increases the number of constraints being processed, so as to fully utilize various infeasible solutions to help find feasible regions. Moreover, a new dynamic resource allocating scheme is proposed to reasonably allocate the limited computational resources to the two populations according to their performance feedback. Experimental results on three test suites and ten practical problems show that the proposed algorithm has a better or competitive performance compared with several state-of-the-art CMOEAs.