Decomposition-based Evolutionary Algorithm Research Articles

Optimal scheduling and management of grid‐connected distributed resources using improved decomposition‐based many‐objective evolutionary algorithm

AbstractThis paper emphasizes the integration of wind and photovoltaic (PV) generation with battery energy storage systems (BESS) in distribution networks (DNs) to enhance grid sustainability, reliability, and flexibility. A novel multi‐objective optimization framework is introduced in this study to minimize energy supply costs, emissions, and energy losses while improving voltage deviation (VD) and voltage stability index (VSI). The proposed framework comprising normal boundary intersection (NBI) and decomposition‐based evolutionary algorithms (DBEA) determines the optimal siting and sizing of renewable‐based distributed resources, considering load demand variations and the intermittency of wind and solar outputs. The comparative analysis establishes that the proposed strategy performs better than many contemporary algorithms, specifically when all the objective functions are optimized simultaneously. The validation of the proposed framework was carried out on the standard IEEE‐33 bus test network, which demonstrates significant percentage savings in energy supply costs (49.6%), emission rate (62.2%), and energy loss (92.3%), along with enormous improvements in VSI (91.9%) and VD (99.8953%). The obtained results categorically underline the efficiency, reliability, and robustness of the proposed approach when employed on any complex distribution network comprising multiple renewable energy sources and battery storage systems.

A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection

Evolutionary algorithms have been widely applied for solving multi-objective optimization problems, while the feature selection in classification can also be treated as a discrete bi-objective optimization problem if attempting to minimize both the classification error and the ratio of selected features. However, traditional multi-objective evolutionary algorithms (MOEAs) may have drawbacks for tackling large-scale feature selection, due to the curse of dimensionality in the decision space. Therefore, in this paper, we concentrated on designing an multi-task decomposition-based evolutionary algorithm (abbreviated as MTDEA), especially for handling high-dimensional bi-objective feature selection in classification. To be more specific, multiple subpopulations related to different evolutionary tasks are separately initialized and then adaptively merged into a single integrated population during the evolution. Moreover, the ideal points for these multi-task subpopulations are dynamically adjusted every generation, in order to achieve different search preferences and evolutionary directions. In the experiments, the proposed MTDEA was compared with seven state-of-the-art MOEAs on 20 high-dimensional classification datasets in terms of three performance indicators, along with using comprehensive Wilcoxon and Friedman tests. It was found that the MTDEA performed the best on most datasets, with a significantly better search ability and promising efficiency.

A problem transformation-based and decomposition-based evolutionary algorithm for large-scale multiobjective optimization

For large-scale multi-objective optimization problems, the search space becomes exceptionally vast, resulting in increased complexity in the search process. The search space usually contains several local optimal individuals, and the difficulty of finding the global optimal individuals increases greatly. A problem transformation-based and decomposition-based large-scale multi-objective evolutionary algorithm is proposed to solve the problem of reducing the dimensionality of the search space and the optimization of populations. Reducing the number of optimization problems in the decision space through problem transformation has lowered the complexity of multi-objective optimization problems and enhanced computational efficiency. In the decision space, employing two direction vectors adaptively guides the generation of promising individuals, thereby preventing the population from falling into local optima. Due to the dimensionality reduction of the decision space and the optimization of the individuals in the objective space, a set of optimal solutions are effectively obtained and uniformly distributed on the approximate Pareto optimal front. Experimental results show that the algorithm is highly competitive with five large-scale multi-objective evolutionary algorithms for large-scale multi-objective optimization test problems with up to 2000 decision variables.

Growing neural gas assisted evolutionary many-objective optimization for handling irregular Pareto fronts

Reference vector adjustment approaches pose a considerable potential to strengthen the capability of decomposition-based evolutionary many-objective algorithms in handling many-objective optimization problems (MaOPs) with irregular Pareto-optimal fronts (PFs). However, most existing approaches often rely on the current population’s distribution, which may not well match MaOPs’ PFs. This can easily mislead the reference vector adjustment and impair algorithms’ performance. To address the abovementioned issue, this paper designs a growing neural gas-assisted evolutionary many-objective optimization algorithm, called GNG-EMO. Specifically, the proposal contains an environmental selection without reference vectors to maintain a well-diversified and converged archive over the PFs. Also, the GNG-EMO leverages a growing neural gas (GNG) network to learn Pareto-optimal fronts’ topologies using the solutions in the archive. In addition, to alleviate the issue that the nodes of the GNG network cannot reach the boundaries of Pareto-optimal fronts, we design an expansion mechanism to further diversify the nodes of the GNG network for constructing a set of better reference vectors. To evaluate the performance of the GNG-EMO, we compare it with five state-of-the-art competitors in the context of 48 test cases with irregular PFs. The numerical results demonstrate the GNG-EMO’s superior competitiveness by significantly outperforming all the five competitors on 27 test cases with respect to the indicator Hypervolume.

A Two-State Dynamic Decomposition-Based Evolutionary Algorithm for Handling Many-Objective Optimization Problems

Decomposition-based many-objective evolutionary algorithms (D-MaOEAs) are brilliant at keeping population diversity for predefined reference vectors or points. However, studies indicate that the performance of an D-MaOEA strongly depends on the similarity between the shape of the reference vectors (points) and that of the PF (a set of Pareto-optimal solutions symbolizing balance among objectives of many-objective optimization problems) of the many-objective problem (MaOP). Generally, MaOPs with expected PFs are not realistic. Consequently, the inevitable weak similarity results in many inactive subspaces, creating huge difficulties for maintaining diversity. To address these issues, we propose a two-state method to judge the decomposition status according to the number of inactive reference vectors. Then, two novel reference vector adjustment strategies, set as parts of the environmental selection approach, are tailored for the two states to delete inactive reference vectors and add new active reference vectors, respectively, in order to ensure that the reference vectors are as close as possible to the PF of the optimization problem. Based on the above strategies and an efficient convergence performance indicator, an active reference vector-based two-state dynamic decomposition-base MaOEA, referred to as ART-DMaOEA, is developed in this paper. Extensive experiments were conducted on ART-DMaOEA and five state-of-the-art MaOEAs on MaF1-MaF9 and WFG1-WFG9, and the comparative results show that ART-DMaOEA has the most competitive overall performance.

Decomposition-based evolutionary algorithm with dual adjustments for many-objective optimization problems

Balancing the convergence and diversity of solutions is a pivotal task for many-objective optimization problems (MaOPs). The decomposition-based evolutionary algorithm has demonstrated great potential in solving MaOPs in the past years. However, its performance degrades when MaOPs have complex Pareto fronts (PFs). Inspired by its pros and cons, this paper proposes a decomposition-based evolutionary algorithm adopting dual adjustments to address MaOPs with irregular PFs. First, an MaOP is divided into a set of subproblems by the distance between weight vectors. Each subproblem selects an appropriate solution from its region, using the specified scalarizing function. Then, the first adjustment updates all the scalarizing functions for each weight vector, where a strategy integrating history information is used to promote the accuracy of the adjustment. Sequentially, the second adjustment updates weight vectors based on the population distribution, which simulates and modifies the value function of reinforcement learning to intensify the rationality of updates. Note that the excitation frequencies of two adjustments are adaptive. Additionally, we design fine-tuning introducing reminding solutions to enhance exploitation. Finally, numerous experiments demonstrate that the proposed algorithm performs better or is equivalent to five state-of-the-art algorithms on 150 test instances and one practical problem . • A novel dual adjustment strategy is proposed, incorporating it into the decomposition-based evolutionary algorithm (DEAdas) to handle the irregular MaOPs. • Adaptive scalarizing functions are presented, and their updates make use of evolutionary information. This adjustment provides an appropriate scalarizing function for each weigh vector. • The adjustment of weight vectors considers the potential of subregions and population distribution, which promotes convergence and diversity. • Experimental results demonstrate that the proposed DEAdas has superiority over peer algorithms.

A many-objective optimization algorithm using a two-space interactive evolutionary framework

The challenge of many-objective optimization evolutionary algorithms with weight vectors is to generate a set of weight vectors that can adapt well to the distribution of the evolving population. In this paper, a many-objective optimization algorithm using a two-space interactive evolutionary framework, called MaO-TIEF, is proposed. In MaO-TIEF, the population space and weight space evolve simultaneously by interacting mutually to achieve the balance of convergence and population diversity in the algorithm. To utilize the useful information in the searching history of individuals, a modified decomposition-based evolutionary algorithm is introduced to update the population space. Moreover, inspired by Steffensen’s method, a local search strategy is adopted to enhance the searching ability of MaO-TIEF. In the experimental section, the proposed algorithm and 5 commonly-used MaOEAs have been tested on 27 test problems with different numbers of objectives and the statistical results demonstrate the effectiveness of MaO-TIEF.

A new adaptive decomposition-based evolutionary algorithm for multi- and many-objective optimization

In decomposition-based multi-objective evolutionary algorithms (MOEAs), a set of uniformly distributed reference vectors (RVs) is usually adopted to decompose a multi-objective optimization problem (MOP) into several single-objective sub-problems, and the RVs are fixed during evolution. When it comes to multi-objective optimization problems (MOPs) with complex Pareto fronts (PFs), the effectiveness of the multi-objective evolutionary algorithm (MOEA) may degrade. To solve this problem, this article proposes an adaptive decomposition-based evolutionary algorithm (ADEA) for both multi- and many-objective optimization. In ADEA, the candidate solutions themselves are used as RVs, so that the RVs can be automatically adjusted to the shape of the Pareto front (PF). Also, the RVs are successively generated one by one, and once a reference vector (RV) is generated, the corresponding sub-objective space is dynamically decomposed into two sub-spaces. Moreover, a variable metric is proposed and merged with the proposed adaptive decomposition approach to assist the selection operation in evolutionary many-objective optimization (EMO). The effectiveness of ADEA is compared with several state-of-the-art MOEAs on a variety of benchmark MOPs with up to 15 objectives. The empirical results demonstrate that ADEA has competitive performance on most of the MOPs used in this study.

A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization

Decomposition-based evolutionary algorithms have recently become fairly popular for many-objective optimization owing to their excellent selection pressure. However, existing decomposition methods are still quite sensitive to the various shapes of the frontiers of many-objective optimization problems (MaOPs). On the one hand, the penalty-based boundary intersection (PBI) method is incapable of acquiring uniform frontiers for MaOPs with very convex frontiers due to the radial distribution of the reference lines. On the other hand, the parallel reference lines of the normal boundary intersection (NBI) method often result in poor diversity for MaOPs with concave frontiers because of under-sampling near the boundaries. In this paper, a collaborative decomposition (CoD) method is first proposed to integrate the advantages of the PBI and NBI methods to overcome their respective disadvantages. This method inherits the NBI-style Tchebycheff function as a convergence measure to improve the convergence and uniformity of the distribution of the PBI method. Moreover, this method also adaptively tunes the extent to which an NBI reference line is rotated towards a PBI reference line for every boundary subproblem to enhance the diversity of the distribution of the NBI method. Furthermore, a CoD-based evolutionary algorithm (CoDEA) is presented for many-objective optimization. A CoD-based ranking strategy is primarily designed in the CoDEA to rank all the individuals associated with every boundary subproblem according to the CoD aggregation function and determine the best ranks. The proposed algorithm is compared with several popular many-objective evolutionary algorithms on 85 benchmark test instances. The experimental results show that the CoDEA achieves high competitiveness, benefiting from the CoD method.

Many-objective optimization of wireless sensor network deployment

Recently, the efficient deployment of wireless sensor networks (WSNs) has become a leading field of research in WSN design optimization. Practical scenarios related to WSN deployment are often considered as optimization models with multiple conflicting objectives that are simultaneously enhanced. In the related literature, it had been shown that moving from mono-objective to multi-objective resolution of WSN deployment is beneficial. However, since the deployment of real-world WSNs encompasses more than three objectives, a multi-objective optimization may harm other deployment criteria that are conflicting with the already considered ones. Thus, our aim is to go further, explore the modeling and the resolution of WSN deployment in a many-objective (i.e., optimization with more than three objectives) fashion and especially, exhibit its added value. In this context, we first propose a many-objective deployment model involving seven conflicting objectives, and then we solve it using an adaptation of the Decomposition-based Evolutionary Algorithm “ $$\theta$$ -DEA”. The developed adaptation is named “WSN- $$\theta$$ -DEA” and is validated through a detailed experimental study.

A Two-Archive Many-Objective Optimization Algorithm Based on D-Domination and Decomposition

Decomposition-based evolutionary algorithms are popular with solving multi-objective optimization problems. It uses weight vectors and aggregate functions to keep the convergence and diversity. However, it is hard to balance diversity and convergence in high-dimensional objective space. In order to discriminate solutions and equilibrate the convergence and diversity in high-dimensional objective space, a two-archive many-objective optimization algorithm based on D-dominance and decomposition (Two Arch-D) is proposed. In Two Arch-D, the method of D-dominance and adaptive strategy adjusting parameter are used to apply selection pressure on the population to identify better solutions. Then, it uses the two archives’ strategy to equilibrate convergence and diversity, and after classifying solutions in convergence archive, the improved Tchebycheff function is used to evaluate the solution set and retain the better solutions. For the diversity archive, the diversity is maintained by making any two solutions as far apart and different as possible. Finally, the Two Arch-D is compared with other four multi-objective evolutionary algorithms on 45 many-objective test problems (including 5, 10 and 15 objectives). Good performance of the algorithm is verified by the description and analysis of the experimental results.

Dynamic Transfer Reference Point-Oriented MOEA/D Involving Local Objective-Space Knowledge

The decomposition-based evolutionary algorithm (MOEA/D) has attained excellent performance in solving optimization problems involving multiple conflicting objectives. However, the Pareto-optimal front (POF) of many multiobjective optimization problems (MOPs) has irregular properties, which weakens the performance of MOEA/D. To address this issue, we devise a dynamic transfer reference point-oriented MOEA/D with local objective-space knowledge (DTR-MOEA/D). The design principle is based on three original and rigorous mechanisms. First, the individuals are projected onto a line segment (two-objective case) or a 3-D plane (three-objective case) after being normalized in the objective space. The line segment or the plane is divided into three different regions: 1) the central region; 2) the middle region; and 3) the edge region. Second, a dynamic transfer criterion of the reference point is developed based on the population density relationships in different regions. Third, a strategy of population diversity enhancement guided by local objective-space knowledge is adopted to improve the diversity of the population. Finally, the experimental results conducted on 16 benchmark MOPs and eight modified MOPs with irregular POF shapes verify that the proposed DTR-MOEA/D has attained a strong competitiveness compared with other representative algorithms.

A decomposition-based evolutionary algorithm using an estimation strategy for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are commonly seen in real-world applications and have attracted a growing attention in recent years. In this paper, a decomposition-based evolutionary algorithm using an estimation strategy is presented to handle MMOPs. In the proposed algorithm, multiple individuals who are assigned to the same weight vector form a subpopulation. Then, the estimation strategy is designed for estimating the amount of Pareto optimal sets (PSs), where the mean-shift algorithm is adopted to classify subpopulation into some clusters. In essence, the number of clusters is considered to be the estimated number of PSs. Finally, an environmental selection method, which combines the estimation strategy and the greedy selection, is adopted to dynamically adjust the subpopulation scale for maintaining the population diversity. The experimental results illustrate that the devised algorithm performs pass beyond the selected up-to-date competing algorithms.

Multi-Objective Evolutionary Algorithm Based on Two Reference Points Decomposition and Historical Information Prediction

Aims: The main goal of this paper is to address the issues of low-quality offspring solutions generated by traditional evolutionary operators, as well as the evolutionary algorithm's inability to solve multi-objective optimization problems (MOPs) with complicated Pareto fronts (PFs). Background: For some complicated multi-objective optimization problems, the effect of the multiobjective evolutionary algorithm based on decomposition (MOEA/D) is poor. For specific complicated problems, there is less research on how to improve the performance of the algorithm by setting and adjusting the direction vector in the decomposition-based evolutionary algorithm. Considering that in the existing algorithms, the optimal solutions are selected according to the selection strategy in the selection stage, without considering whether it could produce the better solutions in the stage of individual generation to achieve the optimization effect faster. As a result, a multi-objective evolutionary algorithm based on two reference points decomposition and historical information prediction is proposed. Objective: In order to verify the feasibility of the proposed strategy, the F-series test function with complicated PFs is used as the test function to simulate the proposed strategy. Method: Firstly, the evolutionary operator based on historical information prediction (EHIP) is used to generate better offspring solutions to improve the convergence of the algorithm; secondly, the decomposition strategy based on ideal point and nadir point is used to select solutions to solve the MOPs with complicated PFs, and the decomposition method with augmentation term is used to improve the population diversity when selecting solutions according to the nadir point. Finally, the proposed algorithm is compared to several popular algorithms by the F-series test function, and the comparison is made according to the corresponding performance metrics. Result: The performance of the algorithm is improved obviously compared with the popular algorithms after using the EHIP. When the decomposition method with augmentation term is added, the performance of the proposed algorithm is better than the algorithm with only the EHIP on the whole, but the overall performance is better than the popular algorithms. Conclusion : The experimental results show that the overall performance of the proposed algorithm is superior to the popular algorithms. The EHIP can produce better quality offspring solutions, and the decomposition strategy based on two reference points can well solve the MOPs with complicated PFs. This paper mainly demonstrates the theory without testing the practical problems. The following research mainly focuses on the application of the proposed algorithm to the practical problems such as robot path planning.

Multifactorial Evolutionary Algorithm Based on Improved Dynamical Decomposition for Many-Objective Optimization Problems

In multiobjective optimization, it is generally known that the boom in computational complexity and search spaces came with a rise in the number of objectives, and this leads to a decrease in selection pressure and the deterioration of the evolutionary process. It follows then that the many-objective optimization problem (MaOP) has become one of the most challenging topics in the field of intelligent optimization. Recently, the multifactorial evolutionary algorithm (MFEA) and its variations, which have shown excellent performance in knowledge transfer across related problems, may provide a new and effective way for solving MaOPs. In this article, a novel MFEA based on improved dynamical decomposition (MFEA/IDD), which integrates the advantages of multitasking optimization and decomposition-based evolutionary algorithms, is proposed. Specifically, in the improved dynamical decomposition strategy (IDD) method, the bi-pivot strategy is designed to provide a good mechanism for balancing between convergence and diversity instead of the single-pivot strategy. Furthermore, a novel MFEA-based approach embedding the IDD strategy is developed to reduce the total running time for solving multiple MaOPs simultaneously. Compared with seven state-of-the-art algorithms, the efficacy of our proposed method is validated experimentally on the benchmarks WFG, DTLZ, and MAF with three to ten objectives, along with a series of real-world cases. The results reveal that the MFEA/IDD is well placed in balancing convergence and diversity while reducing the total number of function evaluations for solving MaOPs.

An integrated framework with evolutionary algorithm for multi-scenario multi-objective optimization problems

Multi-objective optimization problems often load in the multi-scenario environment, and they can be modeled as multi-scenario multi-objective optimization problems (MSMOs). So far, existing research on MSMOs mainly focuses on the specific method in engineering applications, and general methods are quite scarce. This paper develops an integrated information-based evolutionary framework to find a set of scenario-based non-dominated solutions for MSMOs. Sequentially, a scenario-based dominance principle evolutionary algorithm and a decomposition-based evolutionary algorithm are combined with the framework to construct two integration algorithms, respectively. Furthermore, the crucial components in both algorithms are developed under the multi-scenario environment, such as scenario-based constraint handling, scenario-based dominance concept, scenario-based crowded degree and multi-scenario decomposition method. Additionally, we design a novel comprehensive evaluation indicator for MSMOs. Finally, the proposed framework is tested on numerous artificial and engineering problems. Experiments demonstrate that the framework has superiority over four existing state-of-the-art peer competitors concerning the indicator. Two integration algorithms are able to find a set of good scenario-based non-dominated solutions.

A surrogate-assisted evolutionary algorithm for expensive many-objective optimization in the refining process

Computationally expensive optimization problems are difficult for an evolutionary algorithm within limited fitness evaluations, especially for many-objective optimization. To remedy this issue, a surrogate-assisted decomposition-based evolutionary algorithm is proposed in this work where the Kriging model is used to approximate each objective function. In order to balance exploration and exploitation, the model management strategy integrates the infill criterion with the distribution information of both weight vectors and population. Additionally, a one-by-one replacement strategy is developed to select a fixed number of training data to limit computational time for the model training as well as maintain the approximation accuracy. The empirical results show that the performance of the proposed algorithm is competitive to the state-of-the-art algorithms on a number of benchmarks. Finally, the proposed algorithm is applied for a real-world optimization problem in the refining process and exhibits superior performance.

Adaptive Decomposition-Based Evolutionary Algorithm for Many-Objective Optimization with Two-Stage Dual-Density Judgment

Adaptive Decomposition-Based Evolutionary Algorithm for Many-Objective Optimization with Two-Stage Dual-Density Judgment

An Adaptive Reference Vector Adjustment Strategy and Improved Angle-Penalized Value Method for RVEA

Decomposition-based evolutionary multiobjective algorithms (MOEAs) divide a multiobjective problem into several subproblems by using a set of predefined uniformly distributed reference vectors and can achieve good overall performance especially in maintaining population diversity. However, they encounter huge difficulties in addressing problems with irregular Pareto fronts (PFs) since many reference vectors do not work during the searching process. To cope with this problem, this paper aims to improve an existing decomposition-based algorithm called reference vector-guided evolutionary algorithm (RVEA) by designing an adaptive reference vector adjustment strategy. By adding the strategy, the predefined reference vectors will be adjusted according to the distribution of promising solutions with good overall performance and the subspaces in which the PF lies may be further divided to contribute more to the searching process. Besides, the selection pressure with respect to convergence performance posed by RVEA is mainly from the length of normalized objective vectors and the metric is poor in evaluating the convergence performance of a solution with the increase of objective size. Motivated by that, an improved angle-penalized distance (APD) method is developed to better distinguish solutions with sound convergence performance in each subspace. To investigate the performance of the proposed algorithm, extensive experiments are conducted to compare it with 5 state-of-the-art decomposition-based algorithms on 3-, 5-, 8-, and 10-objective MaF1–MaF9. The results demonstrate that the proposed algorithm obtains the best overall performance.

An elitist cooperative evolutionary bi-level multi-objective decomposition-based algorithm for sustainable supply chain

Many real-life applications are modelled using hierarchical decision-making in which: an upper-level optimisation task is constrained by a lower-level one. Such class of optimisation problems is referred in the literature as Bi-Level Optimisation Problems (BLOPs). Most of the proposed methods tackled the single-objective continuous case adhering to some regularity assumptions. This is at odds with real-world problems which involve mainly discrete variables and expensive objective function evaluations. Besides, the optimisation process becomes exorbitantly time-consuming, especially when optimising several objectives at each level. For this reason, the Multi-objective variant (MBLOP) remains relatively less explored and the number of methods tackling the combinatorial case is much reduced. Motivated by these observations, we propose in this work an elitist decomposition-based evolutionary algorithm to solve MBLOPs, called ECODBEMA. The basic idea of our proposal is to handle, decomposition, elitism and multithreading mechanisms to cope with the MBLOP's high complexity. ECODBEMA is applied to the production–distribution problem and to a sustainable end-of-life products disassembly case-study based on real-data of Aix-en-Provence French city. We compared the optimal solutions of an exact method using CPLEX solver with near-optimal solutions obtained by ECODBEMA. The statistical results show the significant outperformance of ECODBEMA against other multi-objective bi-level optimisation algorithms.