Constrained Many-objective Optimization Research Articles

A dual-population Constrained Many-Objective Evolutionary Algorithm based on reference point and angle easing strategy

Constrained many-objective optimization problems (CMaOPs) have gradually emerged in various areas and are significant for this field. These problems often involve intricate Pareto frontiers (PFs) that are both refined and uneven, thereby making their resolution difficult and challenging. Traditional algorithms tend to over prioritize convergence, leading to premature convergence of the decision variables, which greatly reduces the possibility of finding the constrained Pareto frontiers (CPFs). This results in poor overall performance. To tackle this challenge, our solution involves a novel dual-population constrained many-objective evolutionary algorithm based on reference point and angle easing strategy (dCMaOEA-RAE). It relies on a relaxed selection strategy utilizing reference points and angles to facilitate cooperation between dual populations by retaining solutions that may currently perform poorly but contribute positively to the overall optimization process. We are able to guide the population to move to the optimal feasible solution region in a timely manner in order to obtain a series of superior solutions can be obtained. Our proposed algorithm’s competitiveness across all three evaluation indicators was demonstrated through experimental results conducted on 77 test problems. Comparisons with ten other cutting-edge algorithms further validated its efficacy.

A multi-population evolutionary algorithm based on knowledge transfer for constrained many-objective optimization

ABSTRACT Constrained Many-objective Optimization Problems (CMaOPs) are challenging in handling objectives and constraints simultaneously. Here, a novel Constrained Many-objective Optimization Evolutionary Algorithm (CMaOEA) based on Multi-population, Knowledge transfer and Improved environmental selection called CMaMKI is proposed to handle CMaOPs. The proposed framework evolves a task population to solve the original CMaOP and evolves another population to solve a helper problem derived from the original one. To assist solving the original CMaOP, a knowledge expression and transfer strategy is designed to share useful information in the helper population with the task population. Meanwhile, to balance population convergence, diversity and feasibility, an enhanced environmental selection strategy is devised by combining the ε-constrained technique, θ-dominance and subregional density evaluation. The proposed algorithm is evaluated and contrasted with six state-of-the-art algorithms on a set of benchmark CMaOPs. The experimental results demonstrate the superiority and competitiveness of the proposed method.

Solving optimal power flow problems via a constrained many-objective co-evolutionary algorithm

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.

A Constrained Many-Objective Optimization Evolutionary Algorithm With Enhanced Mating and Environmental Selections.

Unlike the considerable research on solving many-objective optimization problems (MaOPs) with evolutionary algorithms (EAs), there has been much less research on constrained MaOPs (CMaOPs). Generally, to effectively solve CMaOPs, an algorithm needs to balance feasibility, convergence, and diversity simultaneously. It is essential for handling CMaOPs yet most of the existing research encounters difficulties. This article proposes a novel constrained many-objective optimization EA with enhanced mating and environmental selections, namely, CMME. It can be featured as: 1) two novel ranking strategies are proposed and used in the mating and environmental selections to enrich feasibility, diversity, and convergence; 2) a novel individual density estimation is designed, and the crowding distance is integrated to promote diversity; and 3) the \θ-dominance is used to strengthen the selection pressure on promoting both the convergence and diversity. The synergy of these components can achieve the goal of balancing feasibility, convergence, and diversity for solving CMaOPs. The proposed CMME is extensively evaluated on 13 CMaOPs and 3 real-world applications. Experimental results demonstrate the superiority and competitiveness of CMME over nine related algorithms.

Handling constrained many-objective optimization problems via determinantal point processes

Although various research studies have been performed to solve constrained multi-objective optimization and many-objective optimization problems, more attention should be paid to the widely existing constrained many-objective optimization problems (CMaOPs). Balancing convergence, diversity and feasibility is an essential issue for CMaOPs. To this end, traditional methods tend to adopt one-by-one selection or deletion strategies. However, these strategies rarely consider the quality of the entirety. Consequently, some poorly performed yet promising solutions might be discarded, finally degrading the performance. To overcome this drawback, this article first introduces the determinantal point processes (DPPs) in Machine Learning to handle CMaOPs. A new algorithm is proposed by applying the DPPs rather than traditional selection strategies for selecting population and archive. To effectively utilize DPPs, two novel Kernel Matrices are designed to represent convergence, diversity, and feasibility qualities of solutions in population and archive, respectively. Besides, a new ε-constrained technique is tailored for the update strategies of archive and corner solution set to handle CMaOPs, especially those with complex constraints. Experiments on 16 different CMaOPs of 77 instances with up to 15 objectives demonstrated the superiority of the proposed method to the state-of-the-art algorithms.

A Dual-Population-Based NSGA-III for Constrained Many-Objective Optimization

The main challenge for constrained many-objective optimization problems (CMaOPs) is how to achieve a balance between feasible and infeasible solutions. Most of the existing constrained many-objective evolutionary algorithms (CMaOEAs) are feasibility-driven, neglecting the maintenance of population convergence and diversity when dealing with conflicting objectives and constraints. This might lead to the population being stuck at some locally optimal or locally feasible regions. To alleviate the above challenges, we proposed a dual-population-based NSGA-III, named DP-NSGA-III, where the two populations exchange information through the offspring. The main population based on the NSGA-III solves CMaOPs and the auxiliary populations with different environment selection ignore the constraints. In addition, we designed an ε-constraint handling method in combination with NSGA-III, aiming to exploit the excellent infeasible solutions in the main population. The proposed DP-NSGA-III is compared with four state-of-the-art CMaOEAs on a series of benchmark problems. The experimental results show that the proposed evolutionary algorithm is highly competitive in solving CMaOPs.

Learning to Optimize: Reference Vector Reinforcement Learning Adaption to Constrained Many-Objective Optimization of Industrial Copper Burdening System.

The performance of decomposition-based algorithms is sensitive to the Pareto front shapes since their reference vectors preset in advance are not always adaptable to various problem characteristics with no a priori knowledge. For this issue, this article proposes an adaptive reference vector reinforcement learning (RVRL) approach to decomposition-based algorithms for industrial copper burdening optimization. The proposed approach involves two main operations, that is: 1) a reinforcement learning (RL) operation and 2) a reference point sampling operation. Given the fact that the states of reference vectors interact with the landscape environment (quite often), the RL operation treats the reference vector adaption process as an RL task, where each reference vector learns from the environmental feedback and selects optimal actions for gradually fitting the problem characteristics. Accordingly, the reference point sampling operation uses estimation-of-distribution learning models to sample new reference points. Finally, the resultant algorithm is applied to handle the proposed industrial copper burdening problem. For this problem, an adaptive penalty function and a soft constraint-based relaxing approach are used to handle complex constraints. Experimental results on both benchmark problems and real-world instances verify the competitiveness and effectiveness of the proposed algorithm.

Two-type weight adjustments in MOEA/D for highly constrained many-objective optimization

A key issue in evolutionary constrained optimization is how to achieve a balance between feasible and infeasible solutions. The quality of generated solutions in decomposition-based multi-objective evolutionary algorithms (MOEAs) depends strongly on the weights’ setting. To fully utilize both the promising feasible and infeasible solutions, this paper proposes two-type weight adjustments based on MOEA/D for solving highly constrained many-objective optimization problems (CMaOPs). During the course of the search, the number of infeasible weights is dynamically reduced, to guide infeasible solutions with better convergence to cross the infeasible barrier, and also to lead infeasible solutions with better diversity to locate multiple feasible subregions. Feasible weights are evenly distributed and keep unchanged throughout the evolution process, which aims to guide the population to search Pareto optimal solutions. The effectiveness of the proposed algorithm is verified by comparing it against six state-of-the-art CMaOEAs on three sets of benchmark problems. Experimental results show that the proposed algorithm outperforms compared algorithms on majority problems, especially on highly constrained optimization problems. Besides, the effectiveness of the proposed algorithm has also been verified on an antenna array synthesis problem.

Hybrid Microgrid Many-Objective Sizing Optimization With Fuzzy Decision

The economics, reliability, and carbon efficiency of hybrid microgrid systems (HMSs) are often in conflict; hence, a reasonable design for the sizing of the initial microgrid is important. In this article, we propose an improved two-archive many-objective evolutionary algorithm (TA-MaEA) based on fuzzy decision to solve the sizing optimization problem for HMSs. For the HMS simulated in this article, costs, loss of power supply probability, pollutant emissions, and power balance are considered as objective functions. For the proposed algorithm, we employ two archives with different diversity selection strategies to balance convergence and diversity in the high-dimensional objective space. In addition, a fuzzy decision making method is proposed to further help decision makers obtain a solution from the Pareto front that optimally balances the objectives. The effectiveness of the proposed algorithm in solving the HMS sizing optimization problem is investigated for the case of Yanbu, Saudi Arabia. The experimental results show that, compared with the two-archive evolutionary algorithm for constrained many-objective optimization (C-TAEA), the clustering-based adaptive many-objective evolutionary algorithm (CA-MOEA), and the improved decomposition-based evolutionary algorithm (I-DBEA), the proposed algorithm can reduce the system costs by 7%, 13%, and 21%, respectively.

A novel hybrid hypervolume indicator and reference vector adaptation strategies based evolutionary algorithm for many-objective optimization

A novel hybrid many-objective evolutionary algorithm called Reference Vector Guided Evolutionary Algorithm based on hypervolume indicator (H-RVEA) is proposed in this paper. The reference vectors are used in a number of sub-problems to decompose the optimization problem. An adaptation strategy is used in the proposed algorithm to adjust the reference vector distribution. The proposed algorithm is compared over well-known benchmark test functions with five state-of-the-art evolutionary algorithms. The results show H-RVEA’s superior performance in terms of the inverted generational distance and hypervolume performance measures than the competitor algorithms. The suggested algorithm’s computational complexity is also analysed. The statistical tests are carried out to demonstrate the statistical significance of the proposed algorithm. In order to demonstrate its efficiency, H-RVEA is also applied to solve two real-life constrained many-objective optimization problems. The experimental results indicate that the proposed algorithm can solve the many-objective real-life problems. Note that the source codes of the proposed technique are available at http://dhimangaurav.com/.

KnRVEA: A hybrid evolutionary algorithm based on knee points and reference vector adaptation strategies for many-objective optimization

In this paper, a many-objective evolutionary algorithm, named as a hybrid evolutionary algorithm based on knee points and reference vector adaptation strategies (KnRVEA) is proposed. Knee point strategy is used to improve the convergence of solution vectors. In the proposed algorithm, a novel knee adaptation strategy is introduced to adjust the distribution of knee points. KnRVEA is compared with five well-known evolutionary algorithms over thirteen benchmark test functions. The results reveal that the proposed algorithm provides better results than the others in terms of Inverted Generational Distance and Hypervolume. The computational complexity of the proposed algorithm is also analyzed. The statistical testing is performed to show the statistical significance of proposed algorithm. The proposed algorithm is also applied on three real-life constrained many-objective optimization problems to demonstrate its efficiency. The experimental results show that the proposed algorithm is able to solve many-objective real-life problems.

Tri-Goal Evolution Framework for Constrained Many-Objective Optimization

It is generally accepted that the essential goal of many-objective optimization is the balance between convergence and diversity. For constrained many-objective optimization problems (CMaOPs), the feasibility of solutions should be considered as well. Then the real challenge of constrained many-objective optimization can be generalized to the balance among convergence, diversity, and feasibility. In this paper, a tri-goal evolution framework is proposed for CMaOPs. The proposed framework carefully designs two indicators for convergence and diversity, respectively, and converts the constraints into the third indicator for feasibility. Since the essential goal of constrained many-objective optimization is to balance convergence, diversity, and feasibility, the philosophy of the proposed framework matches the essential goal of constrained many-objective optimization well. Thus, it is natural to use the proposed framework to deal with CMaOPs. Further, the proposed framework is conceptually simple and easy to instantiate for constrained many-objective optimization. A variety of balance schemes and ranking methods can be used to achieve the balance among convergence, diversity and feasibility. Three typical instantiations of the proposed framework are then designed. Experimental results on a constrained many-objective optimization test suite show that the proposed framework is highly competitive with existing state-of-the-art constrained many-objective evolutionary algorithms for CMaOPs.

An angle based constrained many-objective evolutionary algorithm

Having successfully handled many-objective optimization problems with box constraints only by using VaEA, a vector angle based many-objective evolutionary algorithm in our precursor study, this paper extended VaEA to solve generic constrained many-objective optimization problems. The proposed algorithm (denoted by CVaEA) differs from the original one mainly in the mating selection and the environmental selection, which are made suitable in the presence of infeasible solutions. Furthermore, we suggest a set of new constrained many-objective test problems which have different ranges of function values for all the objectives. Compared with normalized problems, this set of scaled ones is more applicable to test an algorithm’s performance. This is due to the nature property of practical problems being usually far from normalization. The proposed CVaEA was compared with two latest constrained many-objective optimization methods on the proposed test problems with up to 15 objectives, and on a constrained engineering problem from practice. It was shown by the simulation results that CVaEA could find a set of well converged and properly distributed solutions, and, compared with its competitors, obtained a better balance between convergence and diversity. This, and the original VaEA paper, together demonstrate the usefulness and efficiency of vector angle based algorithms for handling both constrained and unconstrained many-objective optimization problems.

A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems.

Many-objective optimization with dynamic constraint handling for constrained optimization problems

In real-world applications, the optimization problems are usually subject to various constraints. To solve constrained optimization problems (COPs), this paper presents a new methodology, which incorporates a dynamic constraint handling mechanism into many-objective evolutionary optimization. Firstly we convert a COP into a dynamic constrained many-objective optimization problem (DCMaOP), which is equivalent to the COP, then the proposed many-objective optimization evolutionary algorithm with dynamic constraint handling, called MaDC, is realized to solve the DCMaOP. MaDC uses the differential evolution (DE) to generate individuals, and a reference-point-based nondominated sorting approach to select individuals. The effectiveness of MaDC is verified on 22 test instances. The experimental results show that MaDC is competitive to several state-of-the-art algorithms, and it has better global search ability than its peer algorithms.

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach

In the precursor paper, a many-objective optimization method (NSGA-III), based on the NSGA-II framework, was suggested and applied to a number of unconstrained test and practical problems with box constraints alone. In this paper, we extend NSGA-III to solve generic constrained many-objective optimization problems. In the process, we also suggest three types of constrained test problems that are scalable to any number of objectives and provide different types of challenges to a many-objective optimizer. A previously suggested MOEA/D algorithm is also extended to solve constrained problems. Results using constrained NSGA-III and constrained MOEA/D show an edge of the former, particularly in solving problems with a large number of objectives. Furthermore, the NSGA-III algorithm is made adaptive in updating and including new reference points on the fly. The resulting adaptive NSGA-III is shown to provide a denser representation of the Pareto-optimal front, compared to the original NSGA-III with an identical computational effort. This, and the original NSGA-III paper, together suggest and amply test a viable evolutionary many-objective optimization algorithm for handling constrained and unconstrained problems. These studies should encourage researchers to use and pay further attention in evolutionary many-objective optimization.