A Coevolutionary Algorithm With Detection and Supervision Strategies for Constrained Multiobjective Optimization

Several constrained optimization problems have been adequately solved over the years thanks to the advances in the area of metaheuristics. Nevertheless, the question as to which search logic performs better on constrained optimization often arises. In this paper, we present Dual Search Optimization (DSO), a co-evolutionary algorithm that includes an adaptive penalty function to handle constrained problems. Compared to other self-adaptive metaheuristics, one of the main advantages of DSO is that it is able auto-construct its own perturbation logics, i.e., the ways solutions are modified to create new ones during the optimization process. This is accomplished by co-evolving the solutions (encoded as vectors of integer/real values) and perturbation strategies (encoded as Genetic Programming trees), in order to adapt the search to the problem. In addition to that, the adaptive penalty function allows the algorithm to handle constraints very effectively, yet with a minor additional algorithmic overhead. We compare DSO with several algorithms from the state-of-the-art on two sets of problems, namely: (1) seven well-known constrained engineering design problems and (2) the CEC 2017 benchmark for constrained optimization. Our results show that DSO can achieve state-of-the-art performances, being capable to automatically adjust its behavior to the problem at hand.

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.

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.

When addressing constrained multi-objective optimization problems, the presence of complex constraints often results in a non-connected feasible region, segmenting the Pareto front into multiple discrete segments. This fragmentation can significantly limit population diversity. To tackle this issue, we have designed two mechanisms aimed at preserving population diversity and have developed a constrained multi-objective co-evolutionary algorithm (DESCA) based on the framework of a two-population co-evolutionary algorithm. The proposed algorithm consists of two populations: a main population dedicated to exploring the constrained Pareto front and an auxiliary population tasked with exploring the unconstrained Pareto front. To sustain the diversity within both populations, the algorithm dynamically adjusts the genetic operator based on the observed states of the populations. Moreover, when the main population encounters stagnation, a regional mating mechanism is employed between the main population and the auxiliary population, accompanied by a relaxation of the constraints on the main population. Conversely, when the auxiliary population experiences stagnation, a diversity-first individual selection strategy is implemented; this strategy utilizes a regional distribution index to assess individual diversity and mitigates population stagnation by enhancing diversity. The performance of DESCA has been evaluated across 33 benchmark problems and 6 real-world problems. Experimental results demonstrate that DESCA exhibits strong competitiveness compared to seven other typical state-of-the-art algorithms.

This paper proposes a new constrained multiobjective optimization algorithm based on artificial immune systems (AIS). To deal with constrained multiobjective optimization problems, the constrained AlS-based multiobjective optimization algorithm is developed by integrating a proposed constraint-handling technique with the unconstrained AIS-based multiobjective optimization algorithm named MOAIS (Xiao and Zu, 2006). We propose the constraint-handling technique by extending a single-objective constraint-handling technique called stochastic ranking (Runarsson and Yao, 2000) to multiobjective optimization process. Two scenarios of the multiobjective version of stochastic ranking are suggested. Thereafter, we develop the constrained MOAIS named MOAIS+SR by integrating the two scenarios with MOAIS. A comparative study is performed quantitatively to assess the performance of MOAIS+SR on a constrained test function suite called CTP test problems. In the comparative study, MOAIS+SR is compared against two other constrained multiobjective algorithms. The simulation results show that the proposed multiobjective stochastic ranking outperforms the constrained-dominance principle (Deb et al., 2000) in handling constraints. Furthermore, we show that the proposed MOAIS+SR achieves the best overall performance among the three algorithms under consideration on the CTP test problems. This study demonstrates that the proposed MOAIS+SR is highly competitive with other state-of-the-art algorithms in constrained multiobjective optimization.

A dual-population auxiliary multiobjective coevolutionary algorithm for constrained multiobjective optimization problems

In real-world applications, the optimization problems usually include some conflicting objectives and subject to many constraints. Much research has been done in the fields of multiobjective optimization and constrained optimization, but little focused on both topics simultaneously. In this study we present a hybrid constraint handling mechanism, which combines the ε-comparison method and penalty method. Unlike original s-comparison method, we set an individual ε-value to each constraint and control it by the amount of violation. The penalty method deals with the region where constraint violation exceeds the ε-value and guides the search toward the ε-feasible region. The proposed algorithm is based on a well-known multiobjective evolutionary algorithm, NSGA-II, and introduces the operators in differential evolution (DE). A modified DE strategy, DE/better-to-best_feasible/l, is applied. The better individual is selected by tournament selection, and the best individual is selected from an archive. Performance of the proposed algorithm is compared with NSGA-II and an improved version with a self-adaptive fitness function. The proposed algorithm shows competitive results on sixteen public constrained multiobjective optimization problem instances.

Many practical problems can be classified as constrained multi-objective optimization problems. Although various methods have been proposed for solving constrained multi-objective optimization problems, there is still a lack of research considering the integration of multiple constraint handling techniques. Given this, this paper combines the objective and constraint separation method with the multi-operator method, proposing a population feasibility state guided autonomous constrained evolutionary optimization method. This method first defines the feasibility state of the population based on both feasibility and ε feasibility of the solutions. Subsequently, a reinforcement learning model is employed to construct a mapping model between the population state and reproduction operators. Finally, based on the real-time population state, the mapping model is utilized to recommend the promising reproduction operator for the next generation. This approach demonstrates significant performance improvement for ε constrained mechanisms in constrained multi-objective optimization algorithms, and shows considerable advantages in comparison with state-of-the-art constrained multi-objective optimization algorithms.

일반적으로 유전자 알고리즘은 최적 시스템을 디자인하는데 주로 이용된다. 하지만 알고리즘의 성능은 적합도 함수나 시스템 환경에 의해 결정된다. 두 개의 개체군이 꾸준히 상호작용하고 공진화 하는 공진화 알고리즘은 이러한 문제를 극복할 수 있을 것으로 기대된다. 본 논문에서는 GA가 풀기 어려운 GA-hard problem을 풀기 위하여 저자가 제안한 3가지 공진화 모델을 설명한다. 첫 번째 모델은 찾고자하는 해와 환경을 각각 경쟁하는 개체군으로 구성해 진화하는 방법으로 사용자의 환경설정에 의해 지역적 해를 찾는 것을 방지하는 경쟁적 공진화 알고리즘이다. 두 번째 모델은 호스트 개체군과 기생(스키마) 개체군으로 구성된 스키마 공진화 알고리즘이다. 이 알고리즘에서 스키마 개체군은 호스트 개체군에 좋은 스키마를 공급한다. 세 번째 알고리즘은 두 개체군이 서로 게임을 통해 진화하도록 하는 게임이론에 기반한 공진화 알고리즘이다. 각 알고리즘은 비주얼 서보잉, 로봇 주행, 다목적 최적화 문제에 적용하여 그 유효성을 입증한다. Usually genetic algorithms are used to design optimal system. However the performance of the algorithm is determined by the fitness function and the system environment. It is expected that a co-evolutionary algorithm, two populations are constantly interact and co-evolve, is one of the solution to overcome these problems. In this paper we propose three types of co-evolutionary algorithm to solve GA-Hard problem. The first model is a competitive co-evolutionary algorithm that solution and environment are competitively co-evolve. This model can prevent the solution from falling in local optima because the environment are also evolve according to the evolution of the solution. The second algorithm is schema co-evolutionary algorithm that has host population and parasite (schema) population. Schema population supply good schema to host population in this algorithm. The third is game model-based co-evolutionary algorithm that two populations are co-evolve through game. Each algorithm is applied to visual servoing, robot navigation, and multi-objective optimization problem to verify the effectiveness of the proposed algorithms.

Many real-world problems involve simultaneous optimization of several incommensurable and often competing objectives. In the search for solutions to multi-objective optimization problems (MOPs), we find that there is no single optimum but rather a set of optimums known as the “Pareto optimal set”. Co-evolutionary algorithms are well suited to optimization problems which involve several often competing objectives. Co-evolutionary algorithms are aimed at evolving individuals through individuals competing in an objective space. In order to approximate the ideal Pareto optimal set, the search capability of diverse individuals in an objective space can be used to determine the performance of evolutionary algorithms. Non-dominated memory and Euclidean distance selection mechanisms for co-evolutionary algorithms have the goal of overcoming the limited search capability of diverse individuals in the population space. In this paper, we propose a method for maintaining population diversity in game model-based co-evolutionary algorithms, and we evaluate the effectiveness of our approach by comparing it with other methods through rigorous experiments on several MOPs.

a new adaptive constrained multi-objective biogeography-based optimization based on two-stage elite selection, ACMBBO, is proposed to solve constrained multi-objective optimization problems. According to the feature on constrained multi-objective and the evolutionary mechanism of BBO, the model of constrained multi-objective optimization which applies to BBO is built. In the model, the habitat suitability index, which combines with the degree of feasible and the Pareto dominance relation between the individuals, is redefined. Moreover, a new mechanism based on two-stage elite selection is set to preserve the elitist of population individuals. Also dynamic migration strategy is designed to improve the ability for exploitation and the utilization of the better individual. Numerical experiments have shown that ACMBBO is competitive with current other constrained multi-objective optimization algorithms on the convergence and the distribution, and is capable of solving the complex CMOPs more effectively and efficiently.

A staged diversity enhancement method for constrained multiobjective evolutionary optimization

The optimal power flow problem in power systems is characterized by a number of complex objectives and constraints, which aim to optimize the total fuel cost, emissions, active power loss, voltage magnitude deviation, and other metrics simultaneously. These conflicting objectives and strict constraints challenge existing optimizers in balancing between active power and reactive power, along with good trade-offs among many metrics. To address these difficulties, this paper develops a co-evolutionary algorithm to solve the constrained many-objective optimization problem of optimal power flow, which evolves three populations with different selection strategies. These populations are evolved towards different parts of the huge objective space divided by large infeasible regions, and the cooperation between them renders assistance to the search for feasible and Pareto-optimal solutions. According to the experimental results on benchmark problems and the IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems, the proposed algorithm is superior over peer algorithms in solving constrained many-objective optimization problems, especially the optimal power flow problems.

Constrained multi-objective optimization assisted by convergence and diversity auxiliary tasks

In this paper, a modular toroid-type SMES was optimized using a recently developed constrained optimization technique named co-evolutionary augmented Lagrangian method (CEALM). The objective of the optimization was to minimize the total length of HTS superconductor satisfying some equality and inequality constraints. The constraints were calculated using 3-D magnetic field analysis techniques and an automatic tetrahedral mesh generator. Optimized results were verified by 3D finite element method (FEM).