An Evolutionary Algorithm With Constraint Relaxation Strategy for Highly Constrained Multiobjective Optimization.

An evolutionary algorithm with directed weights for constrained multi-objective optimization

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: 1) nondominated sorting; 2) reversed nondominated sorting; and 3) 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.

Most of real-world optimization problems correspond to constrained multi-objective optimization problems (CMOPs). Multi-objective evolutionary algorithms (MOEAs) are useful to find out diverse Pareto-optimal solutions in CMOPs, and has been applied in various engineering fields. In order to expand the range of application of MOEAs further, we propose an improved evolutionary algorithm with parallel evaluation strategy (EAPES). In EAPES, feasible solutions and infeasible solution are separately stored in different populations, and infeasible solutions are evaluated in an unusual manner where not only feasible solutions but also useful infeasible solutions are used as parents to reproduce the populations for the next generation. Infeasible solutions are ranked based on a scalarizing function determined by objective function values and a total constraint violation value. This paper investigates the performance of the proposed EAPES to search for Pareto-optimal solutions compared to NSGA-II and the previous EAPES. The proposed EAPES with a well-tuned parameter is most capable to explore Pareto-optimal solutions with good diversity, spread, and convergence to the true Pareto front. The proposed EAPES assigns bad rank to the infeasible solutions that exist away from the true Pareto front, and does not store such solutions. Thus the proposed EAPES exhibits a higher searching capability than the previous EAPES by evaluating infeasible solutions in an appropriate balance between objective functions and total constraint violation. In addition, the results suggest that the proposed EAPES may exhibit high solution search performance even in more difficult CMOPs, which have more objective functions and/or more constraints.

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

Solving constrained multiobjective optimization problems (CMOPs) is highly challenging due to the presence of complicated feasible regions, intense conflicts among objectives, and unevenly distributed constraints. As a result, conventional methods relying on a single constraint-handling mechanism frequently fail to maintain a stable equilibrium among solution feasibility, diversity, and convergence. To overcome these bottlenecks, this article introduces AFFCMO, a novel adaptive feasibility-guided framework tailored for constrained multiobjective optimization. At its core, the proposed approach utilizes a coevolutionary dual-population architecture that divides the search process into two distinct tasks. Specifically, an auxiliary population is tasked with global exploration, while a primary population focuses on the intensive exploitation of discovered feasible areas. To achieve this, the primary population leverages a DE/current-to-pbest/1 differential evolution strategy to closely approximate the constrained Pareto front. Simultaneously, the auxiliary population expands the search space using a mutation operator that adapts to the current evolutionary stage. Furthermore, exploration is bolstered by a multicriterion environmental selection scheme designed for the auxiliary group. By combining Euclidean geometric distributions, constraint relaxation, and value modeling inspired by epidemic dynamics, this strategy successfully preserves valuable infeasible solutions that can guide the search. Additionally, a dynamic resource allocation strategy based on historical search feedback and Thompson sampling is incorporated. This mechanism continuously evaluates the recent search contributions of both populations and adaptively adjusts their offspring sizes, thereby reducing the bias introduced by static allocation schemes. This mechanism continuously assesses the actual search contributions of both populations, allowing for the adaptive resizing of offspring generations and thereby eliminating the inherent biases of static allocation methods. Comprehensive empirical evaluations are conducted on 47 benchmark problems from four distinct test suites. The results indicate that AFFCMO significantly outperforms seven contemporary multiobjective evolutionary algorithms in terms of exploring complex feasible regions, preserving solution diversity, and achieving high convergence accuracy.

Convergence, diversity and feasibility are three important issues when solving constrained multiobjective optimization problems (CMOPs). To deal with the balance among convergence, diversity and feasibility well, this article proposes a cooperative multiobjective evolutionary algorithm with propulsive population (CMOEA-PP) for solving CMOPs. CMOEA-PP has two populations, including propulsive population and normal population, and these two populations work cooperatively. Specifically, propulsive population focuses on convergence. Normal population gives priority to feasibility and is obligated to maintain diversity. To cross through the infeasible region and reach the Pareto front (PF), propulsive population does not consider constraints in the early stage and only considers constraints in the later stage. To further accelerate the speed of convergence, propulsive population only searches for corner solutions and center solutions, while normal population searches for the whole PF. As a result, propulsive population can cross through the infeasible region because of the lack of attention to feasibility. In addition, propulsive population also can guide and accelerate the convergence of the evolutionary process. Comprehensive experiment results on several sets of benchmark problems demonstrate that CMOEA-PP is better than existing state-of-the-art competitors.

Constrained multi-objective optimization problems (CMOPs) appear frequently in engineering applications. In some cases, feasible regions are narrow and/or disconnected. For this kind of problems, existing constraint-handling methods, integrated with multi-objective evolutionary algorithms, are easily stuck at local optima. Aiming to strengthen the global search ability, a dual-grid dual-phase strategy is proposed, which is termed dual-grid push and pull search (DPPS). In the DPPS, two populations, corresponding to dual grids, are used individually to explore the feasible and infeasible spaces. Specifically, one population maintains feasible solutions, and the other explores the whole search space without considering constraints. Then, the two populations share useful information and pull each other so as to enable the algorithm to search for the optimal feasible region (i.e., Pareto solution set). To demonstrate the effectiveness of the proposed algorithm, the MOEA/D integrated DPPS (MOEA/D-DPPS) is tested on a frequently-used benchmark suite as well as a newly-constructed suite. Experimental results clearly show the superiority of MOEA/D-DPPS compared with six state-of-the-art algorithms.

Convergence, diversity and feasibility are difficult to balance in constrained multi-objective optimization problems（CMOP）, especially in some CMOP with relatively small feasibility. In order to solve this problem, this paper proposes a tri-population evolutionary algorithm with multitask co-evolution strategy. The main task is responsible for the original constrained multi-objective optimization. The second task called the help task executes the unconstrained search, and the third task named the transition task fills the gap between the main and the second tasks by treating the degree of constraint violation as another optimization objective. Finally, through the knowledge transfer between the population offspring of the three tasks, the offspring with excellent convergence are transferred to the main task population and get fully exported. In experiments, The algorithm is also compared with four high-level constrained multi-objective evolutionary algorithms (CMOEA) on 24 bench-mark CMOPs. The experimental results demonstrate the competitiveness of the proposed algorithm.

Dual population approximate constrained Pareto front for constrained multiobjective optimization

Constrained multi-objective optimization problems often exhibit complex and fragmented feasible regions, which makes global constrained Pareto front (CPF) search challenging. To address this, we present an unbalanced and balanced competitive strategy-assisted dual-swarm optimizer (UBCSO). UBCSO introduces a balanced swarm that uses random pairing and even grouping to accelerate convergence toward feasible regions, and an unbalanced, cluster-based swarm divides each cluster into three fixed groups for intra-cluster learning, promoting diverse exploration. Within clusters, we propose an unbalanced competition strategy to achieve more precise particle selection during learning, and a cluster-based environmental selection enables effective information exchange across clusters and swarms, enhancing CPF search. Additionally, the proposed cluster integration strategy enables efficient use of computing resources by reducing exploration in non-promising regions. Finally, across 37 instances, UBCSO attained the best overall ranking compared to ten popular methods (IGD+ 1.7838, HV 1.984) and achieved the best IGD+ metric in 25 out of 37 problems.

The effective resolution of constrained multi-objective optimization problems (CMOPs) requires a delicate balance between maximizing objectives and satisfying constraints. Previous studies have demonstrated that multi-swarm optimization models exhibit robust performance in CMOPs; however, their high computational resource demands can hinder convergence efficiency. This article proposes an environment selection model based on Bayes’ theorem, leveraging the advantages of dual populations. The model constructs prior knowledge using objective function values and constraint violation values, and then, it integrates this information to enhance selection processes. By dynamically adjusting the selection of the auxiliary population based on prior knowledge, the algorithm significantly improves its adaptability to various CMOPs. Additionally, a population size adjustment strategy is introduced to mitigate the computational burden of dual populations. By utilizing past prior knowledge to estimate the probability of function value changes, offspring allocation is dynamically adjusted, optimizing resource utilization. This adaptive adjustment prevents unnecessary computational waste during evolution, thereby enhancing both convergence and diversity. To validate the effectiveness of the proposed algorithm, comparative experiments were performed against seven constrained multi-objective optimization algorithms (CMOEAs) across three benchmark test sets and 12 real-world problems. The results show that the proposed algorithm outperforms the others in both convergence and diversity.

A multi-objective evolutionary algorithm for steady-state constrained multi-objective optimization problems

A staged diversity enhancement method for constrained multiobjective evolutionary optimization

Generally, constraint-handling techniques are designed for evolutionary algorithms to solve Constrained Multiobjective Optimization Problems (CMOPs). Most Multiojective Particle Swarm Optimization (MOPSO) designs adopt these existing constraint-handling techniques to deal with CMOPs. In this chapter, the authors present a constrained MOPSO in which the information related to particles' infeasibility and feasibility status is utilized effectively to guide the particles to search for feasible solutions and to improve the quality of the optimal solution found. The updating of personal best archive is based on the particles' Pareto ranks and their constraint violations. The infeasible global best archive is adopted to store infeasible nondominated solutions. The acceleration constants are adjusted depending on the personal bests' and selected global bests' infeasibility and feasibility statuses. The personal bests' feasibility statuses are integrated to estimate the mutation rate in the mutation procedure. The simulation results indicate that the proposed constrained MOPSO is highly competitive in solving selected benchmark problems.

In this paper, a new evolutionary algorithm (EA) to solve multi-objective constrained optimization problem (MCOP) is proposed. First, the rank of the individual and the scalar constraint violation of the individual are defined. Then, based on the rank and the scalar constraint violation of the individual, a new fitness function and a switch selection operator are presented. Accordingly, when the individuals are evaluated or ranked, it doesn't need to care about the feasibility of individuals, therefore it is a penalty-parameterless constraint-handling approach for multi-objective constrained optimization problem. Finally, the computer simulations demonstrate the effectiveness of the proposed algorithm.