For the past many years, several constrained multiobjective evolutionary algorithms (CMOEAs) have been designed for solving constrained multiobjective optimization problems (CMOPs). In these CMOEAs, ...some constraint-handling techniques (CHTs) were proposed to balance the convergence and constrained satisfaction, however, they still face some serious challenges, such as premature convergence to the local optimal region and labor-intensive tuning of parameters for a specific CMOP. Furthermore, most of the existing CHTs are derived by solving constrained single-objective optimization. The information hidden from the feasible nondominated set (FNDS) has not been fully utilized. This study proposed a novel parameter-less constraint handling technique, which divides the entire population into three mutually exclusive subsets dynamically: 1) FNDS; 2) the subset dominated by FNDS; and 3) the subset not dominated by FNDS. According to the proposed division of labor, it is not necessary to balance the convergence and constrained satisfaction in each subset. To avoid being entrapped in local optima, the proposed algorithm adopts a two-stage strategy to solve CMOPs. In the first stage, the proposed algorithm focuses solely on converging toward the unconstrained Pareto front (PF) without considering the constrained satisfaction. In the second stage, the FNDS constraint handling technique is adopted to guide the population converging toward constrained PF effectively. The performance of the proposed algorithm was compared to that of nine state-of-the-art CMOEAs, and the comparison results show that the proposed algorithm performs significantly better on the CF, MW, and LIRCMOP test suites.
•Landscape analysis is extended to constrained multiobjective optimization.•New landscape features are derived to facilitate problem characterization.•Test suites of constrained multiobjective ...optimization problems are thoroughly analyzed.•Artificial test problems fail to fully represent real-world problem characteristics.•The main source of complexity in artificial test problems is violation multimodality.
Despite the increasing interest in constrained multiobjective optimization in recent years, constrained multiobjective optimization problems (CMOPs) are still insufficiently understood and characterized. For this reason, the selection of appropriate CMOPs for benchmarking is difficult and lacks a formal background. We address this issue by extending landscape analysis to constrained multiobjective optimization. By employing four exploratory landscape analysis techniques, we propose 29 landscape features (of which 19 are novel) to characterize CMOPs. These landscape features are then used to compare eight frequently used artificial test suites against a recently proposed suite consisting of real-world problems based on physical models. The experimental results reveal that the artificial test problems fail to adequately represent some realistic characteristics, such as strong negative correlation between the objectives and the overall constraint violation. Moreover, our findings show that all the studied artificial test suites have advantages and limitations, and that no “perfect” suite exists. Additionally, the effectiveness of the proposed features at predicting algorithm performance is demonstrated for two multiobjective optimization algorithms. Benchmark designers can use the obtained results to select or generate appropriate CMOP instances based on the characteristics they want to explore.
When solving constrained multiobjective optimization problems, it is crucial to deal with several conflicting objectives and various constraints simultaneously. To address these two issues, a ...constrained multiobjective differential evolution algorithm with an infeasible-proportion control mechanism is presented in this paper. Specifically, two populations are cooperatively employed in the evolution process. The first population is used to solve the original problem, that is, to explore the constrained Pareto front, while the second population is created to search for high-quality objective function information. Furthermore, differential evolution with two specific mutation strategies is employed to update each population. Cooperative strategies can not only balance diversity and convergence but also realize information exchange between the two populations provide new evolutionary directions. In addition, an infeasible-proportion control mechanism is used to gradually decrease the proportion of infeasible solutions in the second population. This enables the main population to pass through the infeasible region barrier in the early evolution stage and facilitates the search for a constrained Pareto front in the later evolution stage. Systematic experiments on 65 benchmark test functions show that the proposed algorithm is superior to or at least comparable to five well-established constrained multiobjective optimization methods.
The evolutionary algorithm recommendation is catching increasing attention when solving practical application problems since different algorithms often perform differently on different problems. To ...achieve the algorithm recommendation, extracting effective features to accurately characterize the problems is necessary, which is related to the feature extraction problem. So far, most feature extraction methods focus on single-objective optimization problems, and only a few studies are conducted on multiobjective optimization problems and constrained optimization problems, let alone constrained multiobjective optimization problems (CMOPs) that are widely encountered in the real world. To fill the gap, this article proposes an evolution-based constrained multiobjective feature extraction method (ECMOFE), in which the information generated in the evolutionary process is leveraged to form the feature matrix. To be specific, we create two populations to, respectively, optimize constraints and objectives for some generations. Furthermore, two complementary evolutionary operators are used to generate offspring for each population. In the environmental selection, the successful rate of offspring individuals generated by each operator of each population is recorded to form the feature matrix. Then, a dimension reduction method is designed to compress the size of the feature matrix. By the above process, the feature vector that can reflect the global relationship between constraints and objectives and the difficulty of the CMOP is formed. Based on the formed features, several algorithm recommendation methods are built on the basis of classifiers. The results based on multiple metrics show the effectiveness of the proposed ECMOFE.
Among the constraint-handling techniques (CHTs) in constrained multiobjective optimization, constrained dominance principle (CDP) is simple, flexible, nonparametric, and easy to be embedded into ...multiobjective evolutionary algorithms. However, CDP always prefers constraints to objectives, which tends to cause premature convergence. To overcome this drawback, we propose a new CHT on the basis of CDP. In our CHT, the fitness function of each solution is defined as the weighted sum of two rankings: one is the solution's ranking based on CDP and the other is the solution's ranking based on Pareto dominance (i.e., without considering any constraints). It is evident that these two rankings favor the "feasibility" and "optimality" of each solution, respectively. More importantly, the weight employed in the fitness function is related to the proportion of feasible solutions in the current population, enabling the population to adaptively balance constraints and objectives in the evolutionary process. The effectiveness of our CHT is evaluated on three test suites and the experimental results demonstrate that our CHT shows better or highly competitive performance compared with other representative CHTs.
Dynamic constrained multiobjective optimization involves irregular changes in the distribution of the true Pareto-optimal fronts, drastic changes in the feasible region caused by constraints, and the ...movement directions and magnitudes of the optimal distance variables due to diverse changing environments. To solve these problems, we propose a multipopulation evolution-based dynamic constrained multiobjective optimization algorithm. In this algorithm, we design a tribe classification operator to divide the population into different tribes according to a feasibility check and the objective values, which is beneficial for driving the population toward the feasible region and Pareto-optimal fronts. Meanwhile, a population selection strategy is proposed to identify promising solutions from tribes and exploit them to update the population. The optimal values of the distance variables vary differently with dynamic environments, thus, we design a dynamic response strategy for solutions in different tribes that estimates their distances to approach the Pareto-optimal fronts and regenerates a promising population when detecting environmental changes. In addition, a scalable generator is designed to simulate diverse movement directions and magnitudes of the optimal distance variables in real-world problems under dynamic environments, obtaining a set of improved test problems. Experimental results show the effectiveness of test problems, and the proposed algorithm is impressively competitive with several chosen state-of-the-art competitors.
Penalty functions are frequently employed for handling constraints in constrained optimization problems (COPs). In penalty function methods, penalty coefficients balance objective and penalty ...functions. However, finding appropriate penalty coefficients to strike the right balance is often very hard. They are problems dependent. Stochastic Ranking (SR) and constraint-domination principle (CDP) are two promising penalty functions based constraint handling techniques that avoid penalty coefficients. In this paper, the extended/modified versions of SR and CDP are implemented for the first time in the multiobjective evolutionary algorithm based on decomposition (MOEA/D) framework. This led to two new algorithms, CMOEA/D-DE-SR and CMOEA/D-DE-CDP. The performance of these new algorithms is tested on CTP-series and CF-series test instances in terms of the HV-metric, IGD-metric, and SC-metric. The experimental results are compared with NSGA-II, IDEA, and the three best performers of CEC 2009 MOEA competition, which showed better and competitive performance of the proposed algorithms on most test instances of the two test suits. The sensitivity of the performance of proposed algorithms to parameters is also investigated. The experimental results reveal that CDP works better than SR in the MOEA/D framework. Display omitted
► Stochastic ranking (SR) and constraint domination principle (CDP) are studied in the MOEA/D framework to solve CMOPs. ► CDP works better than SR in the MOEA/D framework on most of the CTP-series and CF-series test instances. ► Our algorithms beat IDEA and NSGA-II on five out of eight CTP-series test instances. ► Our algorithms found competitive results with the three best performers in CEC 2009 on six out of ten CF-series test instances. ► Our algorithms can find evenly distributed optimal solutions with a small population size.
Penalty functions are frequently employed for handling constraints in constrained optimization problems (COPs). In penalty function methods, penalty coefficients balance objective and penalty functions. However, finding appropriate penalty coefficients to strike the right balance is often very hard. They are problems dependent. Stochastic ranking (SR) and constraint-domination principle (CDP) are two promising penalty functions based constraint handling techniques that avoid penalty coefficients. In this paper, the extended/modified versions of SR and CDP are implemented for the first time in the multiobjective evolutionary algorithm based on decomposition (MOEA/D) framework. This led to two new algorithms, CMOEA/D-DE-SR and CMOEA/D-DE-CDP. The performance of these new algorithms is tested on CTP-series and CF-series test instances in terms of the HV-metric, IGD-metric, and SC-metric. The experimental results are compared with NSGA-II, IDEA, and the three best performers of CEC 2009 MOEA competition, which showed better and competitive performance of the proposed algorithms on most test instances of the two test suits. The sensitivity of the performance of proposed algorithms to parameters is also investigated. The experimental results reveal that CDP works better than SR in the MOEA/D framework.
This paper develops a neural network architecture and a new processing method for solving in real time, the nonlinear equality constrained multiobjective optimization problem (NECMOP), where several ...nonlinear objective functions must be optimized in a conflicting situation. In this processing method, the NECMOP is converted to an equivalent scalar optimization problem (SOP). The SOP is then decomposed into several-separable subproblems processable in parallel and in a reasonable time by multiplexing switched capacitor circuits. The approach which we propose makes use of a decomposition-coordination principle that allows nonlinearity to be treated at a local level and where coordination is achieved through the use of Lagrange multipliers. The modularity and the regularity of the neural networks architecture herein proposed make it suitable for very large scale integration implementation. An application to the resolution of a physical problem is given to show that the approach used here possesses some advantages of the point of algorithmic view, and provides processes of resolution often simpler than the usual techniques.
The significant issue to solve constrained multiobjective optimization problems (CMOPs) is to keep the balance between objectives and constraints, but the existing evolutionary algorithms are in the ...face of challenges for solving CMOPs with complex feasible regions. Based on this purpose, a constrained multiobjective evolutionary algorithm with the two-archive weak cooperation (CMOEA-TWC) is proposed in this article. In CMOEA-TWC, two archives are evolved, which are denoted as driving archive and normal archive, respectively. Besides, the weak cooperation of two archives is designed for sharing valuable information between archives. Specifically, the driving archive only considers objectives, while the normal archive considers both objectives and constraints. In addition, the minimum shift-based density estimation-based (SDE) distance is adopted to enhance the diversity of solutions in the driving archive, and a strict constrained dominance principle is designed to improve the feasibility of solutions in the normal archive. The proposed algorithm is tested on 47 CMOPs of 4 benchmark suites with the comparison of 4 state-of-the-art algorithms. The experimental results demonstrate that the average ranks in terms of inverted generational distance (IGD) outperform those of existing state-of-the-art competitors.
Constrained multiobjective optimization problems widely exist in real-world applications. To handle them, the balance between constraints and objectives is crucial, but remains challenging due to ...non-negligible impacts of problem types. In our context, the problem types refer particularly to those determined by the relationship between the constrained Pareto-optimal front (PF) and the unconstrained PF. Unfortunately, there has been little awareness on how to achieve this balance when faced with different types of problems. In this article, we propose a new constraint handling technique (CHT) by taking into account potential problem types. Specifically, inspired by the prior work, problems are classified into three primary types: 1) I; 2) II; and 3) III, with the constrained PF being made up of the entire, part and none of the unconstrained counterpart, respectively. Clearly, any problem must be one of the three types. For each possible type, there exists a tailored mechanism being used to handle the relationships between constraints and objectives (i.e., constraint priority, objective priority, or the switch between them). It is worth mentioning that exact problem types are not required because we just consider their possibilities in the new CHT. Conceptually, we show that the new CHT can make a tradeoff among different types of problems. This argument is confirmed by experimental studies performed on 38 benchmark problems, whose types are known, and a real-world problem (with unknown types) in search-based software engineering. Results demonstrate that within both decomposition-based and nondecomposition-based frameworks, the new CHT can indeed achieve a good tradeoff among different problem types, being better than several state-of-the-art CHTs.
