•A novel CHT is proposed based on Pareto dominance-based ranking and CDP-based ranking.•A search algorithm based on four mutation operators is proposed.•A new constrained multiobjective differential ...evolution algorithm is proposed.
The tradeoff between objective functions and constraints is a key issue that needs to be addressed by constrained multiobjective optimization algorithms, and constraint handling techniques (CHTs) are an important technique for balancing objective functions and constraints. In this paper, a novel CHT that fuses two rankings is proposed. Specifically, each individual is assigned two rankings: one ranking calculated based on Pareto dominance (regardless of constraints) and another calculated based on the constrained dominance principle (CDP). The fitness value of an individual is the weighted sum of these two rankings, and the weight is related to the generation number and the proportion of feasible solutions in the current generation. Based on the proposed CHT, a constrained multiobjective differential evolution algorithm is proposed. To generate high-quality offspring, the proposed constrained multiobjective differential evolution algorithm combines four mutation operations as core components of the search algorithm. The proposed algorithm is compared with eight state-of-the-art algorithms in experiments with five test suites, and the experimental results show that the proposed algorithm performs significantly better than the eight state-of-the-art algorithms.
Balancing objectives and constraints is challenging in addressing constrained multiobjective optimization problems (CMOPs). Existing methods may have limitations in handling various CMOPs due to the ...complex geometries of the Pareto front (PF). And the complexity arises from the constraints that narrow the feasible region. Categorizing problems based on their geometric characteristics facilitates facing this challenge. For this purpose, this article proposes a novel constrained multiobjective optimization framework with detection and supervision phases, called COEA-DAS. The framework categorizes the problems into four types based on the overlap between the obtained approximate unconstrained PF and constrained PF to guide the coevolution of the two populations. In the detection phase, the detection population approaches the unconstrained PF ignoring the constraints. The main population is guided by the detection population to cross infeasible barriers and approximate the constrained PF. In the supervision phase, specialized evolutionary mechanisms are designed for each possible problem type. The detection population maintains evolution to assist the main population in spreading along the constrained PF. Meanwhile, the supervision strategy is conducted to reevaluate the problem types based on the evolutionary state of the populations. This idea of balancing constraints and objectives based on the type of problem provides a novel approach for more effectively addressing the CMOPs. Experimental results indicate that the proposed algorithm performs better or more competitively on 57 benchmark problems and 12 real-world CMOPs compared with eight state-of-the-art algorithms.
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.
This article presents a new constraint-handling technique (CHT), called shift-based penalty (ShiP), for solving constrained multiobjective optimization problems. In ShiP, infeasible solutions are ...first shifted according to the distributions of their neighboring feasible solutions. The degree of shift is adaptively controlled by the proportion of feasible solutions in the current parent and offspring populations. Then, the shifted infeasible solutions are penalized based on their constraint violations. This two-step process can encourage infeasible solutions to approach/enter the feasible region from diverse directions in the early stage of evolution, and guide diverse feasible solutions toward the Pareto optimal solutions in the later stage of evolution. Moreover, ShiP can achieve an adaptive transition from both diversity and feasibility in the early stage of evolution to both diversity and convergence in the later stage of evolution. ShiP is flexible and can be embedded into three well-known multiobjective optimization frameworks. Experiments on benchmark test problems demonstrate that ShiP is highly competitive with other representative CHTs. Further, based on ShiP, we propose an archive-assisted constrained multiobjective evolutionary algorithm (CMOEA), called ShiP+, which outperforms two other state-of-the-art CMOEAs. Finally, ShiP is applied to the vehicle scheduling of the urban bus line successfully.
To promote research on dynamic constrained multiobjective optimization, we first propose a group of generic test problems with challenging characteristics, including different modes of the true ...Pareto front (e.g., convexity-concavity and connectedness-disconnectedness) and the changing feasible region. Subsequently, motivated by the challenges presented by dynamism and constraints, we design a dynamic constrained multiobjective optimization algorithm with a nondominated solution selection operator, a mating selection strategy, a population selection operator, a change detection method, and a change response strategy. The designed nondominated solution selection operator can obtain a nondominated population with diversity when the environment changes. The mating selection strategy and population selection operator can adaptively handle infeasible solutions. If a change is detected, the proposed change response strategy reuses some portion of the old solutions in combination with randomly generated solutions to reinitialize the population, and a steady-state update method is designed to improve the retained previous solutions. The experimental results show that the proposed test problems can be used to clearly distinguish the performance of algorithms, and that the proposed algorithm is very competitive for solving dynamic constrained multiobjective optimization problems in comparison with state-of-the-art algorithms.
For constrained multiobjective optimization problems (CMOPs), the ultimate goal is to obtain a set of well-converged and well-distributed feasible solutions to approximate the constrained Pareto ...front (CPF). Various constraints may change the position and/or shape of the CPF. This poses great challenges to the approximation of the CPF. This is especially true when the CPF mainly lies on constraint boundaries (i.e., CPF and unconstrained PF have little or even no intersection). To tackle this issue, we propose a novel dual population algorithm for approximating the CPF from both sides of the constraint boundaries. Specifically, Population1 uses the constrained-domination principle to approximate the CPF from the sides of feasible regions only; Population2 adopts an improved ϵ-constrained method to approximate the CPF from both the feasible as well as infeasible regions. Offspring generated by both populations are merged and combined with Population1 and Population2. In addition, some selected members of Population1 and Population2 are permitted to migrate to the combined populations to facilitate knowledge sharing. Systematic experiments carried out on three benchmark test suites and 10 real-world CMOPs show the proposed algorithm achieved superior or competitive performance, especially for CMOPs where the CPF is mainly located at constraint boundaries. Therefore, on the basis of dual population, approximating CPFs from both sides of feasible and infeasible regions contributes an alternative approach to solving CMOPs.
•A novel dual population algorithm for approximating the CPF from both sides of the constraint boundaries is designed.•Selected members are permitted to migrate to the combined populations to facilitate knowledge sharing.•DPACMO achieved superior performance, especially for CMOPs where the CPF is mainly located at constraint boundaries.
In addition to the search for feasible solutions, the utilization of informative infeasible solutions is important for solving constrained multiobjective optimization problems (CMOPs). However, most ...of the existing constrained multiobjective evolutionary algorithms (CMOEAs) cannot effectively explore and exploit those solutions and, therefore, exhibit poor performance when facing problems with large infeasible regions. To address the issue, this article proposes a novel method, called DD-CMOEA, which features dual stages (i.e., exploration and exploitation) and dual populations. Specifically, the two populations, called mainPop and auxPop, first individually evolve with and without considering the constraints, responsible for exploring feasible and infeasible solutions, respectively. Then, in the exploitation stage, mainPop provides information about the location of feasible regions, which facilitates auxPop to find and exploit surrounding infeasible solutions. The promising infeasible solutions obtained by auxPop in turn help mainPop converge better toward the Pareto-optimal front. Extensive experiments on three well-known test suites and a real-world case study fully demonstrate that DD-CMOEA is more competitive than five state-of-the-art CMOEAs.
The existing constrained multiobjective evolutionary algorithms (CMOEAs) still have great room for improvement in balancing populations convergence, diversity and feasibility on complex constrained ...multiobjective optimization problems (CMOPs). Besides, their effectiveness deteriorates dramatically when facing the CMOPs with scaling-up objective space or search space. We are thus motivated to design a learning-aided CMOEA with promising problem-solving ability and scalability for various CMOPs. In the proposed solver, two learning models are respectively trained online on constrained-ignored task and feasibility-first task, which are then used to learn the two improvement-based vectors for enhancing the search by differential evolution. In addition, the union population of parent and child solutions is divided into multiple subsets with a hierarchical clustering based on cosine similarity. A comprehensive indicator, considering objective-based performance and constraint violation degree of a solution, is developed to select the representative solution from each cluster. The effectiveness of the proposed optimizer is verified by solving the CMOPs with various irregular Pareto fronts, the number of objectives ranging from 2 to 15, and the dimensionality of search space scaling up to 1000.
Constrained multiobjective optimization problems (CMOPs) involve both conflicting objective functions and various constraints. Due to the presence of constraints, CMOPs' Pareto-optimal solutions are ...very likely lying on constraint boundaries. The experience from the constrained single-objective optimization has shown that to quickly obtain such an optimal solution, the search should surround the boundary of the feasible region from both the feasible and infeasible sides. In this article, we extend this idea to cope with CMOPs and, accordingly, we propose a novel constrained multiobjective evolutionary algorithm with bi directional co evolution, called BiCo. BiCo maintains two populations, that is: 1) the main population and 2) the archive population. To update the main population, the constraint-domination principle is equipped with an NSGA-II variant to move the population into the feasible region and then to guide the population toward the Pareto front (PF) from the feasible side of the search space. While for updating the archive population, a nondominated sorting procedure and an angle-based selection scheme are conducted in sequence to drive the population toward the PF within the infeasible region while maintaining good diversity. As a result, BiCo can get close to the PF from two complementary directions. In addition, to coordinate the interaction between the main and archive populations, in BiCo, a restricted mating selection mechanism is developed to choose appropriate mating parents. Comprehensive experiments have been conducted on three sets of CMOP benchmark functions and six real-world CMOPs. The experimental results suggest that BiCo can obtain quite competitive performance in comparison to eight state-of-the-art-constrained multiobjective evolutionary optimizers.
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