Paired Offspring Generation for Constrained Large-Scale Multiobjective Optimization

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.

Highly constrained multiobjective optimization problems (HCMOPs) refer to constrained multiobjective optimization problems (CMOPs) with complex constraints and small feasible regions, which are commonly encountered in many real-world applications. Current constraint-handling techniques will face two difficulties when dealing with HCMOPs: 1) feasible solution is hard to be found and too much search effort is spent in locating the feasible region and 2) since the total feasible region of an HCMOP can consist of several disconnected subregions, the search process might be stuck in the comparatively larger feasible subregion, which does not contain the whole Pareto front (PF). To address these two issues, an evolutionary algorithm with constraint relaxation strategy based on differential evolution algorithm, that is, CRS-DE, is proposed in this article. In each generation, the CRS-DE relaxes the constraints by dividing the infeasible solutions into two subpopulations based on total constraint violation, that is, the ``semifeasible'' subpopulation (SF) and ``infeasible'' subpopulation (IF), respectively. The SF provides information on the promising regions of finding the feasible solution and is the driving force for convergence toward the PF, while the IF focuses on global exploration for new promising regions. Corresponding reproduction and selection strategies are devised for the SF, IF, and feasible subpopulations, which create a clear division of labor with cooperation to facilitate the search for feasible solutions. To leverage the influence of CRS and prevent the population from premature convergence, a mobility restriction mechanism is developed to restrict the individuals in the SF and IF from entering the feasible subpopulation and enhance the diversity of the whole population. Comprehensive experiments on a series of benchmark test problems and a real-world CMOP demonstrate the competitiveness of our method compared with other representative algorithms in terms of effectiveness and reliability in finding a set of well-distributed optimal solutions for HCMOPs.

A dual-population algorithm based on alternative evolution and degeneration for solving constrained multi-objective optimization problems

A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization

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

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

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.

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

Immunity-based hybrid evolutionary algorithm for multi-objective optimization in global container repositioning

Push and pull search embedded in an M2M framework for solving constrained multi-objective optimization problems

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.

The elements of the population of the artificial physics optimization (APO) algorithm are assigned mass, velocity, and displacement attributes. It is a new type of heuristic algorithm, but it still has defects such as low efficiency of non-dominated solution selection and unbalanced search capacity of global and local. This paper introduces the mechanism of improved fast non-dominant sorting and partition-guided individual evolution into the APO algorithm to overcome this imperfection. An improved multi-objective artificial physics optimization algorithm based on fast non-dominated sorting (IFNS-MOAPO) is proposed, which is the result of integrating numerous strategies into the APO algorithm. Firstly, an improved fast non-dominated sorting strategy is introduced. This strategy can increase the efficiency of selecting non-dominated solutions and decrease the running time of the algorithm. Secondly, the mechanism of individual evolution guided by partition is proposed. For individuals in infeasible and feasible domains, different mass functions and virtual force calculation rules are adopted to update the algorithm iteratively to boost the convergence performance. To verify the comprehensive performance of the IFNS-MOAPO algorithm, seven benchmark test problems are selected for simulation experiments and compared with five algorithms in terms of runtime duration, Pareto front plots, and metric values. The results show that the IFNS-MOAPO algorithm has good distribution and can converge to the true Pareto front quickly. It is a useful tool for solving constrained multi-objective optimization problems (CMOPs).

Solving constrained multi-objective optimization problems (CMOPs) is one of the most popular research topics in the multi-objective optimization community. Various approaches based on different algorithmic strategies have been proposed for benchmark CMOPs with different features and challenges. Although most of these algorithms employ one or more fixed strategies, determining the most suitable strategy is challenging and problem-dependent. This work presents a general multitasking framework comprising a main task for the original CMOP and an arbitrary number of auxiliary tasks based on different strategies for the auxiliary problems. Reinforcement learning techniques are utilized according to the population state to choose the most suitable auxiliary task during the evolutionary process. As instantiations, Q-Learning and Deep Q-Learning are successfully applied in the framework. Accordingly, two novel algorithms are developed to solve CMOPs. The results demonstrate the effectiveness of the proposed approaches on five benchmark CMOP test suites and real-world problems compared to 11 advanced methods.

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.

In this paper, we design a set of multi-objective constrained optimization problems (MCOPs) and propose a new repair operator to address them. The proposed repair operator is used to fix the solutions that violate the box constraints. More specifically, it employs a reversed correction strategy that can effectively avoid the population falling into local optimum. In addition, we integrate the proposed repair operator into two classical multi-objective evolutionary algorithms MOEA/D and NSGA-II. The proposed repair operator is compared with other two kinds of commonly used repair operators on benchmark problems CTPs and MCOPs. The experiment results demonstrate that our proposed approach is very effective in terms of convergence and diversity.