Constrained Multi-objective Optimization Problem Research Articles

A spherical vector-based adaptive evolutionary particle swarm optimization for UAV path planning under threat conditions

Unmanned aerial vehicle (UAV) path planning is a constrained multi-objective optimization problem. With the increasing scale of UAV applications, finding an efficient and safe path in complex real-world environments is crucial. However, existing particle swarm optimization (PSO) algorithms struggle with these problems as they fail to consider UAV dynamics, resulting in many infeasible solutions and poor convergence to optimal solutions. To address these challenges, we propose a spherical vector-based adaptive evolutionary particle swarm optimization (SAEPSO) algorithm. This algorithm, based on spherical vectors, directly incorporates UAV dynamic constraints and introduces improved tent map and reverse learning to enhance the diversity and distribution of initial solutions. Additionally, dynamic nonlinear and adaptive factors are integrated to balance exploration and exploitation capabilities. To avoid local optima in highly complex environments, we propose an adaptive acceleration strategy for poor particles, and an evolutionary programming strategy is incorporated to further improve the optimization capability. Finally, we conducted comparative studies and in six benchmark scenarios with varying threat levels, and the results demonstrated that the proposed algorithm outperforms others in the initial solution effectiveness, the final solution accuracy, convergence stability, and scalability.

Two-stage bidirectional coevolutionary algorithm for constrained multi-objective optimization

Objective optimization and constraint satisfaction are two primary and conflicting tasks in solving constrained multi-objective optimization problems (CMOPs). To better trade off them, this paper proposes a two-stage bidirectional coevolutionary algorithm, termed C-TBCEA, for constrained multi-objective optimization. It consists of two stages, with each concentrating on specific targets, i.e., the first stage primarily focuses on objective optimization while the second stage focuses on constraint satisfaction by employing different evolutionary strategies at each stage. Via the synergy of the two stages, a dynamic trade-off between objective optimization and constraint satisfaction can be achieved, thus overcoming the distinctive challenges that may be encountered at different stages of evolution. In addition, to take advantage of both feasible and infeasible solutions, we employ two populations, i.e., the main population that stores the non-dominated feasible solutions and the archive population that maintains the informative infeasible solutions, to prompt the bidirectional coevolution of them. To validate the effectiveness of the proposed C-TBCEA, experiments are carried out on 6 CMOP test suites and 17 real-world CMOPs. The results demonstrate that the proposed algorithm is very competitive with 9 state-of-the-art constrained multi-objective optimization evolutionary algorithms (CMOEAs).

Deep reinforcement learning-guided coevolutionary algorithm for constrained multiobjective optimization

Effectively managing convergence, diversity, and feasibility constitutes a fundamental trinity of tasks in optimizing constrained multiobjective optimization problems (CMOPs). Nevertheless, contemporary constrained multiobjective evolutionary algorithms (CMOEAs) frequently encounter challenges in reconciling these imperatives simultaneously. Drawing inspiration from overwhelming success in artificial intelligence, we propose a deep reinforcement learning-guided coevolutionary algorithm (DRLCEA) to tackle this predicament. DRLCEA employs two populations to optimize the original and unconstrained versions of the CMOP, respectively and then fosters cooperation between them according to the guidance of DRL. The established DRL employs two evaluation metrics to appraise population convergence, diversity, and feasibility, thus remarkably proficient in reflecting and steering the coevolution. Therefore, the proposed DRLCEA could effectively locate the feasible regions and approximate the constrained Pareto front. We assess the proposed algorithm on 32 benchmark CMOPs and one real-world UAV emergency track planning (UETP) application. Experimental results undoubtedly demonstrate the superiority and robustness of the proposed DRLCEA.

A multi-objective partitioning algorithm for large-scale graph based on NSGA-II

Graph partitioning plays a crucial role in distributed graph computation. The existing algorithms often formulate the partition problem as a single objective optimization problem focusing on load balancing and minimizing communication overhead, which is difficult to achieve optimization for multiple objectives and neglects communication balance. This paper formulates the problem of graph partitioning as a multi-objective constrained optimization problem, and proposes an enhanced NSGA-II to solve it. The objectives include achieving load balance, minimizing the communication overhead, and keeping communication balance. After partitioning the graph according to the vertices’ degrees in first, the NSGA-II combined neighbor search is applied to guide the vertices migration to obtain the best partition result. Compared with Hash, GAP, HLP, CP_WSG, and SPNL using dataset from SNAP, the experiments performed on a cluster of 4 computing nodes demonstrate that the load balance ratio and cut-edge ratio of the proposed algorithm is almost the same with those of the best existing algorithm, and it can achieve the better cut-edge balance ratio and communication balance ratio than the other algorithms.

Voting enabled dynamic constraint portfolios for evolutionary multi-objective optimization

Constraint handling technique (CHT) plays a key role in handling constrained multi-objective optimization problems (CMOPs). In general, CHTs are associated with their own pros and cons under different scenarios, giving rise to an inherent risk in the selection of appropriate CHTs. To take advantages of diverse CHTs, we design a novel competitive co-evolution scheme for CHT portfolio in which multiple complementary CHTs are synergized via a single-population competitive ensemble framework. A self-organizing voting strategy is developed to allocate restrictive resources to constituent CHTs adaptively according to the feedbacks of CHTs as the search proceeds. In addition, an archive update strategy is designed to facilitate effective usage of potential solutions and achieve better compromise between convergence and diversity. Then the concrete algorithm, namely, VP-CMOEA, is presented. We have comprehensively evaluated our proposal on diverse benchmark functions and real-world applications. Empirical results demonstrate the better generality of VP-CMOEA compared with state-of-the-art peers.

Optimal siting and sizing of battery energy storage systems in unbalanced distribution systems: A multi objective problem under uncertainty

In this paper the siting and sizing problem of battery energy storage systems in unbalanced active distribution systems is formulated as a mixed-integer, non-linear, constrained multi objective (MO) optimization problem under uncertainties. The problem is cumbersome from the computational point of view due to the presence of intertemporal constraints, a great number of state variables and the presence of uncertainties in the problem input data. A new approach based on the trade-off/risk analysis is proposed to obtain with acceptable computational efforts a solution that may not be the optimal solution but represents a reasonable and robust compromise. We use the trade-off/risk analysis, because it was specifically developed for power system planning problems in which we deal with a wide range of options, with possible conflicting objectives, and with uncertainty and risk. The proposed approach includes new procedures to select an adequate set of planning alternatives to be considered in the trade-off/risk analysis framework and to assist the planning engineer when difficulties arise in setting probabilities of the input data. Numerical applications to an IEEE unbalanced test system demonstrate the effectiveness of the proposed procedure and indicate the best alternatives of storage systems in the range from 450 kW to 600 kW globally installed in a reduced set of nodes.

A staged fuzzy evolutionary algorithm for constrained large-scale multiobjective optimization

Constrained multiobjective optimization problems (CMOPs) are prevalent in practical applications, where constraints play a significant role. Building on techniques from constrained single-objective optimization, classical methods such as the constrained dominance principle have been extended to CMOPs. However, these methods struggle with CMOPs characterized by complex infeasible regions. Furthermore, as the number of decision variables increases, the search efficiency of algorithms deteriorates dramatically. To solve those issues, we propose a staged fuzzy evolutionary algorithm (i.e., SFEA) for constrained large-scale problems. To balance exploration and exploitation, a fuzzy stage adjustment strategy based on the sigmoid function is proposed. Furthermore, this article develops an improved fuzzy operator to perform fuzzy operations on various vectors (e.g., solutions or constraint violations). Computational experiments were conducted on CMOP test suites with up to 500 decision variables and a series of real-world applications. The experimental results demonstrate that, compared to existing peer algorithms, our algorithm exhibits superior or competitive performance.

An adaptive co-evolutionary competitive particle swarm optimizer for constrained multi-objective optimization problems

In constrained multi-objective optimization problems, it is challenging to balance the convergence, diversity and feasibility of the population, especially encountering complex infeasible regions. In order to effectively balance the three indicators, from the aspects of the handling of infeasible solution and the quality of individuals, a multi-population co-evolutionary competitive particle swarm optimization algorithm hybridized with infeasible solution transfer and an adaptive technique (ACCPSO) is proposed. Firstly, the information of feasible and infeasible individuals is fully utilized and the individuals are classified by Hamming distance. Then, a novel constraint handling technique based on learning from the promising feasible direction is designed to make individuals cross large infeasible regions and explore more potential feasible regions. Moreover, aiming to provide robust search capability and consequently further generate high-quality solutions, the genetic operators and the particle swarm optimization operator with the competitive mechanism are introduced as operators with an adaptive mechanism. Finally, compared with the state-of-the-art methods, the performance of the proposed algorithm is verified on LIR-CMOP, MW and DTLZ, as well as two real-world problems. The results indicate that ACCPSO exhibits stronger competitiveness in terms of convergence, the solution quality, and distribution diversity on the feasible Pareto front.

A comparative study of evolutionary algorithms and particle swarm optimization approaches for constrained multi-objective optimization problems

Many real-world optimization problems contain multiple conflicting objectives as well as additional problem constraints. These problems are referred to as constrained multi-objective optimization problems (CMOPs). Many meta-heuristics for solving CMOPs, called constrained multi-objective meta-heuristics (CMOMHs) have been introduced in the literature, including those using particle swarm optimization (PSO)(Kennedy and Eberhart, 1995), genetic algorithms (GAs)(Man et al., 1996), and differential evolution (DE)(Storn and Price, 1997). CMOMHs can be grouped into four different classes: classic CMOMHs, co-evolutionary approaches, multi-stage approaches, and multi-tasking approaches. An extensive comparative study of twenty different CMOMHs on a wide variety of test problems, including real-world CMOPs in the fields of science and engineering, is conducted. A multi-swarm PSO approach called constrained multi-guide particle swarm optimization (ConMGPSO) is introduced and compared to the best-performing previous approaches according to the comparative study. The performance of each algorithm was found to be problem dependent, however the best overall approaches were ConMGPSO, paired-offspring constrained evolutionary algorithm (POCEA)(He et al., 2021), adaptive non-dominated sorting genetic algorithm III (A-NSGA-III)(Jain and Deb, 2014), and constrained multi-objective framework using Q-learning and evolutionary multi-tasking (CMOQLMT)(Ming and Gong, 2023). ConMGPSO and POCEA had the best performance on the CF benchmark set, which contains examples of bi-objective and tri-objective CMOPs with disconnected CPOFs. The CMOQLMT approach had the best performance on the DAS-CMOP benchmark set, which contain additional difficulty in terms of feasibility-, convergence-, and diversity-hardness. For the selected real-world CMOPs, A-NSGA-III had the best performance overall. ConMGPSO was shown to have the best performance on the process, design, and synthesis problems, and had competitive performance for the power system optimization problems.

Adaptive Constraint Relaxation-Based Evolutionary Algorithm for Constrained Multi-Objective Optimization

The key problem to solving constrained multi-objective optimization problems (CMOPs) is how to achieve a balance between objectives and constraints. Unfortunately, most existing methods for CMOPs still cannot achieve the above balance. To this end, this paper proposes an adaptive constraint relaxation-based evolutionary algorithm (ACREA) for CMOPs. ACREA adaptively relaxes the constraints according to the iteration information of population, whose purpose is to induce infeasible solutions to transform into feasible ones and thus improve the ability to explore the unknown regions. Completely ignoring constraints can cause the population to waste significant resources searching for infeasible solutions, while excessively satisfying constraints can trap the population in local optima. Therefore, balancing constraints and objectives is a crucial approach to improving algorithm performance. By appropriately relaxing the constraints, it induces infeasible solutions to be transformed into feasible ones, thus obtaining more information from infeasible solutions. At the same time, it also establishes an archive for the storage and update of solutions. In the archive update process, a diversity-based ranking is proposed to improve the convergence speed of the algorithm. In the selection process of the mating pool, common density selection metrics are incorporated to enable the algorithm to obtain higher-quality solutions. The experimental results show that the proposed ACREA algorithm not only achieved the best Inverse Generation Distance (IGD) value in 54.6% of the 44 benchmark test problems and the best Hyper Volume (HV) value in 50% of them, but also obtained the best results in seven out of nine real-world problems. Clearly, CP-TSEA outperforms its competitors.

An Improved Coevolutionary Algorithm for Constrained Multi-Objective Optimization Problems

Constrained multi-objective optimization problems are ubiquitous in engineering applications. In recent years, constrained multi-objective optimization algorithms based on the dual population coevolutionary framework have been widely studied due to their excellent performance. However, when facing optimization problems with complex constraints, the performance of existing algorithms still needs further improvement. This paper proposes an improved constrained multi-objective coevolutionary algorithm (iCMOCA). The algorithm mainly includes two populations: One population takes into account constraints, while the other population disregards them. Meanwhile, the iCMOCA employs effective collaboration between two populations during the process of offspring generation and environmental selection, and it utilizes an environmental selection strategy based on multi-objective to multi-objective decomposition to improve the performance. Comparative analysis conducted on the DAS-CMOP and MW test suites provides empirical evidence that iCMOCA outperforms five state-of-the-art algorithms.

Deep reinforcement learning assisted novelty search in Voronoi regions for constrained multi-objective optimization

Solving constrained multi-objective optimization problems (CMOPs) requires optimizing multiple conflicting objectives while satisfying various constraints. Existing constrained multi-objective evolutionary algorithms (CMOEAs) cross infeasible regions by ignoring constraints. However, these methods might neglect promising search directions, leading to insufficient exploration of the search space. To address this issue, this paper proposes a deep reinforcement learning assisted constrained multi-objective quality-diversity algorithm. The proposed algorithm designs a diversity maintenance mechanism to promote evenly coverage of the final solution set on the constrained Pareto front. Specifically, first, a novelty-oriented archive is created using a centroid Voronoi tessellation, which divides the search space into a desired number of Voronoi regions. Each region acts as a repository of non-dominated solutions with different phenotypic characteristics to provide diversity information and supplementary evolutionary trails. Secondly, to improve resource utilization, a deep Q-network is adopted to learn a policy to select suitable Voronoi regions for offspring generation based on their novelty scores. The exploration of these regions aims to find a set of diverse, high-performing solutions to accelerate convergence and escape local optima. Compared with eight state-of-the-art CMOEAs, experimental studies on four benchmark suites and nine real-world applications demonstrate that the proposed algorithm exhibits superior or at least competitive performance, especially on problems with discrete and narrow feasible regions.

Constrained large-scale multiobjective optimization based on a competitive and cooperative swarm optimizer

Many engineering application problems can be modeled as constrained multiobjective optimization problems (CMOPs), which have attracted much attention. In solving CMOPs, existing algorithms encounter difficulties in balancing conflicting objectives and constraints. Worse still, the performance of the algorithms deteriorates drastically when the size of the decision variables scales up. To address these issues, this study proposes a competitive and cooperative swarm optimizer for large-scale CMOPs. To balance conflict objectives and constraints, a bidirectional search mechanism based on competitive and cooperative swarms is designed. It involves two swarms, approximating the true Pareto front from two directions. To enhance the search efficiency in large-scale space, we propose a fast-converging competitive swarm optimizer. Unlike existing competitive swarm optimizers, the proposed optimizer updates the velocity and position of all particles at each iteration. Additionally, to reduce the search range of the decision space, a fuzzy decision variables operator is used. Comparison experiments have been performed on test instances with 100–1000 decision variables. Experiments demonstrate the superior performance of the proposed algorithm over five peer algorithms.

Surrogate-assisted push and pull search for expensive constrained multi-objective optimization problems

In many real-world engineering optimizations, a large number of objective and constraint function values often need to be obtained through simulation software or physical experiments, which incurs significant computational costs and/or time expenses. These problems are known as expensive constraint multi-objective optimization problems (ECMOPs). This paper combines the push and pull search (PPS) framework and proposes a surrogate-assisted evolutionary algorithm to solve ECMOPs through Bayesian active learning, naming it the surrogate-assisted PPS (SA-PPS). Specifically, during the push search stage, candidate solutions are selected based on two indicators: hypervolume improvement and objective uncertainty. These aim to quickly guide the population towards the unconstrained Pareto front while ensuring diversity. During the pull search stage, the population is partitioned into many subregions through reference vectors, and different selection strategies are assigned to each subregion based on its state, aiming to guide the population towards the constrained Pareto front while ensuring diversity. Furthermore, we introduce a batch data selection strategy that utilizes Bayesian active learning to enable the surrogate model to focus on regions of interest in the pull search stage. Extensive experimental results have shown that the proposed SA-PPS algorithm exhibits superior convergence and diversity compared to 9 state-of-the-art algorithms across a variety of benchmark problems and a real-world optimization problem.

Interval Constrained Multi-Objective Optimization Scheduling Method for Island-Integrated Energy Systems Based on Meta-Learning and Enhanced Proximal Policy Optimization

Multiple uncertainties from source–load and energy conversion significantly impact the real-time dispatch of an island integrated energy system (IIES). This paper addresses the day-ahead scheduling problems of IIES under these conditions, aiming to minimize daily economic costs and maximize the output of renewable energies. We introduce an innovative algorithm for Interval Constrained Multi-objective Optimization Problems (ICMOPs), which incorporates meta-learning and an improved Proximal Policy Optimization with Clipped Objective (PPO-CLIP) approach. This algorithm fills a notable gap in the application of DRL to complex ICMOPs within the field. Initially, the multi-objective problem is decomposed into several single-objective problems using a uniform weight decomposition method. A meta-model trained via meta-learning enables fine-tuning to adapt solutions for subsidiary problems once the initial training is complete. Additionally, we enhance the PPO-CLIP framework with a novel strategy that integrates probability shifts and Generalized Advantage Estimation (GAE). In the final stage of scheduling plan selection, a technique for identifying interval turning points is employed to choose the optimal plan from the Pareto solution set. The results demonstrate that the method not only secures excellent scheduling solutions in complex environments through its robust generalization capabilities but also shows significant improvements over interval-constrained multi-objective evolutionary algorithms, such as IP-MOEA, ICMOABC, and IMOMA-II, across multiple multi-objective evaluation metrics including hypervolume (HV), runtime, and uncertainty.

Coevolutionary multitasking for constrained multiobjective optimization

Addressing the challenges of constrained multiobjective optimization problems (CMOPs) with evolutionary algorithms requires balancing constraint satisfaction and optimization objectives. Coevolutionary multitasking (CEMT) offers a promising strategy by leveraging synergies from distinct, complementary tasks. The primary challenge in CEMT frameworks is constructing suitable auxiliary tasks that effectively complement the main CMOP task. In this paper, we propose an adaptive CEMT framework (ACEMT), which customizes two adaptive auxiliary tasks to enhance CMOP-solving efficiency. The first auxiliary task dynamically narrows constraint boundaries, facilitating exploration in regions with smaller feasible spaces. The second task focuses specifically on individual constraints, continuously adapting to expedite convergence and uncover optimal regions. In solving the main CMOP task, this dual-auxiliary-task strategy not only improves search thoroughness but also clarifies the balance between constraints and objectives. Concretely, ACEMT incorporates an adaptive constraint relaxation technique for the first auxiliary task and a specialized constraint selection strategy for the second. These innovations foster effective knowledge transfer and task synergy, addressing the key challenge of auxiliary task construction in CEMT frameworks. Extensive experiments on three benchmark suites and real-world applications demonstrate ACEMT’s superior performance compared to state-of-the-art constrained evolutionary algorithms. ACEMT sets a new standard in CMOP-solving by strategically constructing and adapting auxiliary tasks, representing a significant advancement in this research direction.

Manifold-assisted coevolutionary algorithm for constrained multi-objective optimization

In constrained multi-objective optimization problems (CMOPs), constraints often fragment the Pareto solution space into multiple feasible and infeasible regions. This fragmentation presents a challenge for evolutionary optimization methods as feasible regions can be discrete and isolated by infeasible areas, making exploration difficult and leading to populations getting trapped in local optima. To address these issues, this paper introduces a manifold assisted coevolutionary algorithm for solving CMOPs. Firstly, a guided feasible search strategy is proposed to explore feasible regions, especially those isolated by infeasible barriers. This is achieved by estimating directions to the Constrained Pareto Set (CPS). Secondly, a manifold learning-based exploration strategy is employed to spread the population along the Pareto Set (PS) manifold by estimating the manifold distribution. Moreover, two populations are exploited, where the first population serves as the primary population, considering both constraints and objectives to explore the feasible region and search along the CPS. The second population, on the other hand, does not consider constraints and serves as an auxiliary population to explore the Unconstrained PS. By cooperating, these two populations effectively approach and cover separated CPS segments. The proposed algorithm is evaluated against seven state-of-the-art algorithms on 37 CMOP test functions and 5 CMOPs with fraudulent constraints. The experimental results clearly demonstrate that our algorithm can reliably locate multiple CPSs and is considered state-of-the-art in handling CMOPs.

Second-order Karush/Kuhn–Tucker conditions and duality for constrained multiobjective optimization problems

In this paper, we investigate the second-order strong Karush/Kuhn–Tucker conditions and duality of a constrained multiobjective optimization problem (CMOP). Exploiting a second-order regularization condition, we obtain second-order strong KKT necessary conditions of Borwein-properly efficient solution of CMOP without convexity assumptions. Further, second-order sufficient conditions for the second-order KKT point to be an efficient solution of CMOP are derived under the generalized second-order convexity assumptions. Finally, we establish duality results between CMOP and its second-order Wolfe-type dual problem.

EvolutionViT: Multi-objective evolutionary vision transformer pruning under resource constraints

Vision Transformer (ViT) has emerged as a pivotal model for a variety of visual tasks, surpassing convolutional neural networks by a substantial margin. However, the performance of ViT is seriously impaired by intensive computational and storage costs requirements, posing significant barriers for real-world applications or deployment on resource-constrained edge devices. To address this limitation, compressing the ViT to accelerate its inference at no appreciable degradation of vision performance has attracted widespread attention. Although there are some studies on accelerating ViT, they seldom consider resource constraints and multi-criteria decision making in the process. This article formulates ViT pruning as a large-scale constrained multi-objective optimization problem, and proposes a patch pruning framework for accelerating ViT, called EvolutionViT, based on the developed multi-objective optimization model. EvolutionViT can effectively tradeoff between computational cost and performance under resource constraints, automatically searching for solutions while optimizing two conflicting objectives. In particular, exploiting the knee solution and boundary solutions to directly guide the entire evolutionary process, EvolutionViT can efficiently identify a knee solution that satisfies the resource constraints, which in turn avoids the manual search for a good trade-off. To verify and evaluate our proposed method, we compare EvolutionViT with a number of representative ViT models on the ImageNet dataset. The comprehensive simulation results show that the proposed EvolutionViT demonstrates a competitive advantage compared to peers, with significantly reduced computational expense at the cost of slightly degraded performance.

A constrained multi-objective evolutionary algorithm based on fitness landscape indicator

Constrained multi-objective optimization problems (CMOPs) with constraints in both the decision and objective space are shown to be great challenges to be solved. Considering the different requirements of different problems on resource allocation of exploration and exploitation, this paper proposes a new constrained multi-objective evolutionary algorithm based on a novel fitness landscape indicator. The indicator regards the fitness landscape and evolutionary generation among the population to determine the selection of the offspring generation mechanism. The proposed algorithm uses the new indicator to select different differential evolutions during the evolutionary process to balance exploration and exploitation. Numerical experiments on three test suites and three practical examples compared with six existing algorithms show the proposed algorithm can effectively deal with different types of CMOPs, especially in CMOPs with constraints in both the decision and objective spaces.