Large-scale Global Optimization Research Articles

An Efficient Differential Grouping Algorithm for Large-Scale Global Optimization

Cooperative co-evolution (CC) is a practical and efficient evolutionary framework for solving large-scale global optimization problems (LSGOPs). The performance of CC depends on how variables are being grouped and can be improved through guided variable decomposition for various optimization problems. However, achieving a proper variable decomposition is computationally expensive. This paper proposes an effective yet efficient differential grouping (EDG) method to reduce the associated computational cost. Our method exploits the historical interrelationship information of previous variable groups to examine interactions between the remnant variable groups. This allows us to spend less computing resources without compromising the accuracy of the final grouping result. Our proposal utilizes the covariance matrix adaptation evolution strategy (CMA-ES) algorithm, in conjunction with EDG, to solve LSGOPs. Further, to reduce time complexity and improve the stability of CMA-ES, we substitute the complex matrix decomposition step with simpler matrix operations to compute the square root of the covariance matrix. Results from our experiments and analysis indicate that EDG is a competitive method to solve LSGOPs and improve the performance of CC. The proposed schemes significantly enhance the searchability of CMA-ES compared to the other large-scale variants of CMA-ES and state-of-the-art large-scale optimizers. Moreover, our EDG could be integrated with evolutionary optimizers of different flavors like Differential Evolution (DE).

An effective theoretical and experimental analysis method for the improved slime mould algorithm

Metaheuristic intelligent optimization algorithms are effective methods for solving high-dimensional nonlinear complex optimization problems. The slime mould algorithm is a novel intelligent optimization algorithm proposed in 2020. However, the basic slime mould algorithm still has some shortcomings, such as slow convergence rate, easy to fall into local extremum, and unbalanced exploration and development capability. To further improve and expand the optimization ability and application scope of the slime mould algorithm, and enhance its performance in solving large-scale complex optimization problems, this paper proposes a slime mould algorithm (PPMSMA) based on Gaussian perturbation and phased position update, positive variation, and multi-strategy greedy selection. Firstly, Gaussian perturbation and phased position update mechanism are introduced to avoid the difficulty of the algorithm to jump out of the local extrema and also to speed up the convergence of the algorithm. Then, a positive variation strategy based on the sine cosine mechanism is introduced to move the variation of the population towards a better direction. Finally, a multi-strategy greedy selection mechanism is introduced, which effectively improves the search ability of the algorithm. The analysis and research on the optimization ability and performance of metaheuristic algorithms mainly include two aspects: theoretical analysis and experimental testing. Theoretical analysis has always been a relatively weak link in the research of metaheuristic algorithms, and there is currently no clear and effective method formed. For experimental testing, although there are more methods, they often lack systematization and adequacy. In this paper, a more complete, fine-grained and systematic approach to theoretical and experimental analysis is proposed. In the theoretical analysis part, the time complexity and spatial complexity of the PPMSMA algorithm are analytically proved to be the same as the basic slime mould algorithm, and the probability measure method is used to prove that PPMSMA algorithm can converge to the global optimal solution. In the simulation experiment section, the PPMSMA algorithm is compared with multiple sets of 10 representative comparison algorithms on the CEC2017 complex test function set suite for optimization accuracy analysis, Friedman comprehensive ranking analysis, average optimization rate analysis of PPMSMA relative to other algorithms, convergence curve analysis, and Wilcoxon rank-sum test analysis. To further examine the scalability of the improved algorithm in solving large-scale optimization problems, PPMSMA is compared with the above 10 comparative algorithms under 1000 dimensional conditions in the large-scale global optimization test set CEC2010, and the solution stability of each algorithm is analyzed through violin plots. The results show that the PPMSMA algorithm has significantly improved convergence performance, optimization accuracy, and solution stability in both high-dimensional and large-scale complex problems, and has significant advantages compared to multiple sets of 10 representative comparative algorithms. Finally, PPMSMA and 10 other comparative algorithms are used to solve engineering design optimization problems with different complexities. The experimental results validate the universality, reliability, and superiority of PPMSMA in handling engineering design constraint optimization problems.

Large-scale evolutionary optimization: A review and comparative study

Large-scale global optimization (LSGO) problems have widely appeared in various real-world applications. However, their inherent complexity, coupled with the curse of dimensionality, makes them challenging to solve. Continuous efforts have been devoted to designing computational intelligence-based approaches to solve them. This paper offers a comprehensive review of the latest developments in the field, focusing on the advances in both single-objective and multi-objective large-scale evolutionary optimization algorithms over the past five years. We systematically categorize these algorithms, discuss their distinct features, and highlight benchmark test suites essential for performance evaluation. After that, comparative studies are conducted using numerical solutions to evaluate the performance of state-of-the-art LSGO for both single-objective and multi-objective problems. Finally, we discuss the real-world applications of LSGO, some challenges, and possible future research directions.

Using cooperative coevolution in large-scale black-box constraint satisfaction problems

Solving constrained large-scale global optimization problems poses a challenging task. In these problems with constraints, when the number of variables is measured in the thousands, when the constraints are presented in the form of a black box, and neither the size nor the configuration of the feasible region is known, it is very difficult to find at least one feasible solution. In general, such a problem of finding a feasible region is known as a constraint satisfaction problem. In this paper, we have extended a well-known benchmark set based on constrained optimization problems up to 1000 variables. We have evaluated the CC-SHADE performance, to tackle constraints in large-scale search space. CC-SHADE merges the power of cooperative coevolution and self-adaptive differential evolution. Our extensive experimental evaluations on a range of benchmark problems demonstrate the strong dependence of the performance of CC-SHADE on the number of individuals and the subcomponent number. The numerical results emphasize the importance of using a cooperative coevolution framework for evolutionary-based approaches compared to conventional methods. All numerical experiments are proven by the Wilcoxon test.

A survey of meta-heuristic algorithms in optimization of space scale expansion

Optimization problem of space scale expansion widely exists in practical applications, such as transportation, logistics, scheduling, social networks, etc. According to different expansion directions, the problem of space scale expansion can be divided into three categories: expansion of decision space, expansion of objective space and simultaneous expansion of decision space and objective space. These three types of problems correspond to large-scale global optimization problem, multi/many-objective optimization problem, and large-scale multi/many-objective optimization problem, respectively. Driven by the above problems, meta-heuristic algorithms (MHAs) with scalable characteristics have received extensive attention in this field. This paper summarizes the research progress of MHAs for space scale expansion optimization from three perspectives. Starting from the key difficulties of the optimization problem, the challenges brought by the expansion of the space scale to the existing MHAs are emphatically analyzed. From the perspective of methodology, the optimization methods of MHAs are divided into three modules: simplify problem structure, improve algorithm performance and expand application fields. Based on the review of performance evaluation benchmark problems, the simulation degree of various test suites for different practical application problems is summarized. In addition, some remaining challenges and future research directions on optimization of space scale expansion are discussed and analyzed.

A Decomposition Method for Both Additively and Nonadditively Separable Problems

Problem decomposition is crucial for coping with large-scale global optimization problems, which relies heavily on highly precise variable grouping methods. The state-of-the-art decomposition methods identify separability based on the finite differences principle, which is valid only for additively separable functions but not applicable to non-additively separable functions. Therefore, we need to investigate separability in more depth in order to propose a more general principle and design more universal decomposition methods. In this paper, we conduct a comprehensive theoretical investigation on separability, the core of which is proposing an innovative separability identification principle: the minimum points shift principle. By utilizing the new principle, we develop a general separability grouping (GSG) method that can handle both additively and non-additively separable functions with high accuracy. In addition, we design a new set of benchmark functions based on non-additive separability, which compensates for the lack of non-additively separable functions in the previous test suites. Extensive experiments demonstrate that the proposed GSG achieves high grouping accuracy on both new and CEC series benchmark problems, especially on non-additively separable problems Finally, we verify that the proposed GSG can effectively improve the optimization performance of non-additively separable problems through optimization experiments.

Dynamical Sphere Regrouping Particle Swarm Optimization: A Proposed Algorithm for Dealing with PSO Premature Convergence in Large-Scale Global Optimization

Optimizing large-scale numerical problems is a significant challenge with numerous real-world applications. The optimization process is complex due to the multi-dimensional search spaces and possesses several locally optimal regions. In response to this issue, various metaheuristic algorithms and variations have been developed, including evolutionary and swarm intelligence algorithms and hybrids of different artificial intelligence techniques. Previous studies have shown that swarm intelligence algorithms like PSO perform poorly in high-dimensional spaces, even with algorithms focused on reducing the search space. However, we propose a modified version of the PSO algorithm called Dynamical Sphere Regrouping PSO (DSRegPSO) to avoid stagnation in local optimal regions. DSRegPSO is based on the PSO algorithm and modifies inertial behavior with a regrouping dynamical sphere mechanism and a momentum conservation physics effect. These behaviors maintain the swarm’s diversity and regulate the exploration and exploitation of the search space while avoiding stagnation in optimal local regions. The DSRegPSO mechanisms mimic the behavior of birds, moving particles similar to birds when they look for a new food source. Additionally, the momentum conservation effect mimics how birds react to collisions with the boundaries in their search space or when they are looking for food. We evaluated DSRegPSO by testing 15 optimizing functions with up to 1000 dimensions of the CEC’13 benchmark, a standard for evaluating Large-Scale Global Optimization used in Congress on Evolutionary Computation, and several journals. Our proposal improves the behavior of all variants of PSO registered in the toolkit of comparison for CEC’13 and obtains the best result in the non-separable functions against all the algorithms.

Low-Dimensional Space Modeling-Based Differential Evolution for Large-Scale Global Optimization Problems

Large-Scale Global Optimization (LSGO) has been an active research field. Part of this interest is supported by its application to cutting-edge research such as Deep Learning, Big Data, and complex real-world problems such as image encryption, real-time traffic management, and more. However, the high dimensionality makes solving LSGO a significant challenge. Some recent research deal with the high dimensionality by mapping the optimization process to a reduced alternative space. Nonetheless, these works suffer from the changes in the search space topology and the loss of information caused by the dimensionality reduction. This paper proposes a hybrid metaheuristic, so-called LSMDE (Low-dimensional Space Modeling-based Differential Evolution), that uses the Singular Value Decomposition to build a low-dimensional search space from the features of candidate solutions generated by a new SHADE-based algorithm (GM-SHADE). GM-SHADE combines a Gaussian Mixture Model (GMM) and two specialized local algorithms: MTS-LS1 and L-BFGS-B, to promote a better exploration of the reduced search space. GMM mitigates the loss of information in mapping high-dimensional individuals to low-dimensional individuals. Furthermore, the proposal does not require prior knowledge of the search space topology, which makes it more flexible and adaptable to different LSGO problems. The results indicate that LSMDE is the most efficient method to deal with partially separable functions compared to other state-of-the-art algorithms and has the best overall performance in two of the three proposed experiments. Experimental results also show that the new approach achieves competitive results for non-separable and overlapping functions on the most recent test suite for LSGO problems.

An Adaptive Dimension Weighting Spherical Evolution to Solve Continuous Optimization Problems

The spherical evolution algorithm (SE) is a unique algorithm proposed in recent years and widely applied to new energy optimization problems with notable achievements. However, the existing improvements based on SE are deemed insufficient due to the challenges arising from the multiple choices of operators and the utilization of a spherical search method. In this paper, we introduce an enhancement method that incorporates weights in individuals’ dimensions that are affected by individual fitness during the iteration process, aiming to improve SE by adaptively balancing the tradeoff between exploitation and exploration during convergence. This is achieved by reducing the randomness of dimension selection and enhancing the retention of historical information in the iterative process of the algorithm. This new SE improvement algorithm is named DWSE. To evaluate the effectiveness of DWSE, in this study, we apply it to the CEC2017 standard test set, the CEC2013 large-scale global optimization test set, and 22 real-world problems from CEC2011. The experimental results substantiate the effectiveness of DWSE in achieving improvement.

Contribution-Based Cooperative Co-Evolution With Adaptive Population Diversity for Large-Scale Global Optimization [Research Frontier

Contribution-Based Cooperative Co-Evolution With Adaptive Population Diversity for Large-Scale Global Optimization [Research Frontier

Improved dwarf mongoose optimization algorithm using novel nonlinear control and exploration strategies

The Dwarf Mongoose Optimization Algorithm (DMO) is a popular metaheuristic algorithm utilized to solve real-world problems. However, DMO has limitations such as slow convergence rate, susceptibility to local optima, and poor performance in high-dimensional problems. Aiming at the defects of the DMO, we propose an improved version called IDMO. This paper introduces the best leader and proposes a novel nonlinear control strategy based on the sine function, significantly enhancing the convergence rate while ensuring accuracy. Moreover, we propose a novel exploration strategy to solve the global optimization and high-dimensional problem. To demonstrate the performance of the proposed IDMO, we test on 65 test functions, including CEC2017, CEC2020, CEC2022, and CEC2013 large-scale global optimization. IDMO is compared with three classes of widely recognized algorithms: (1) highly-cited algorithms, namely, GSA, GWO, WOA, and HHO, (2) advanced algorithms, such as CPSOGSA, CSA, AVOA, and SO, (3) high-performance optimizers including L-SHADE, AL-SHADE, LSHADE-SPACMA, and LSHADE-cnEpSin. Moreover, we also compare with the original DMO and variants such as GDNNMOA, CO-DWO, and BDMOSAO. Meanwhile, we apply IDMO to solve 19 engineering design problems from the CEC2020 real-world optimization suite. Experimental results demonstrate that IDMO outperforms other algorithms, exhibiting excellent convergence rate, stability, and accuracy. The effectiveness of IDMO is confirmed by the Friedman mean ranking, showcasing its potential for metaheuristic optimization and real-world engineering design problems.

Enhanced opposition-based grey wolf optimizer for global optimization and engineering design problems

A recently developed swarm-based meta-heuristic algorithm namely Grey Wolf Optimization algorithm (GWO), which is based on the hunting and leadership behaviours of the grey wolves in nature, has shown superior performance when compared with existing meta-heuristic algorithms. However, like other approaches, the GWO has the limitation of poor exploitation ability and being stuck in local optima when solving challenging optimization problems. To overcome these limitations, a novel technique, namely “Enhanced Opposition-Based Learning” (EOBL), has been proposed and is implemented with the GWO algorithm. The EOBL technique is largely inspired by Opposition-Based Learning (OBL) and Random Opposition-Based Learning (ROBL) techniques to efficiently balance exploration and exploitation. As a result, the Enhanced Opposition-Based Grey Wolf Optimizer (EOBGWO), an innovative approach, is proposed to increase the effectiveness of the conventional GWO algorithm. To test the efficiency of the proposed EOBGWO method, it has been tested on the standard IEEECEC2005, IEEECEC2017, and IEEECEC2019 test functions, along with several real-life engineering design problems. Furthermore, to evaluate the effectiveness and stability of the proposed technique, it has been evaluated on the challenging IEEECEC2008 special session on large scale global optimization problems. The experimental outcomes and statistical measures such as the t-test and Wilcoxon rank-sum test demonstrate that the proposed EOBGWO method outperforms the other state-of-the-art algorithms in both global optimization and engineering design problems.

A space-reduction based three-phase approach for large-scale optimization

With the rapid development of science and technology, large-scale global optimization (LSGO) has received more and more attention. However, large-scale problem is very challenging due to its high dimensionality, nonlinearity, and countless local optimal solutions. In this paper, a space-reduction based three-phase approach (SRTP) is proposed to solve this problem. In the first phase, a space-reduction based line search method is designed to roughly locate good solutions as well as to explore important information such as promising search regions and characteristics of each variable (dimension). In the second phase, a multi-grouping strategy is proposed for fully non-separable large-scale problems. This new decomposition method provides dozens of grouping results for non-separable large-scale problems that not only reduces the dimensionality but also maintains the correlation of variables to the most extent. Then in the third phase, a space-reduction based group search method is designed to optimize the sub-groups. By decomposing the large-scale problem as well as reducing the huge search space to focus on promising areas, this group search method can gain better efficiency. Experiments are conducted on 35 widely used benchmark functions and the proposed SRTP is compared with state-of-the-art LSGO algorithms. Experimental results show that SRTP is effective and efficient.

Spark-based cooperative coevolution for large scale global optimization

Spark-based cooperative coevolution for large scale global optimization

A cultural evolution with a modified selection function and adaptive [formula omitted]-cognition procedure for numerical optimization

In recent years, several population-based evolutionary and swarm algorithms have been developed and used in the literature. This work introduces an improved Cultural Algorithm with a modified selection function and a dynamic α-cognition procedure to handle a variety of challenging numerical optimization problems. The modified selection function is used to support a balanced evolutionary search. A process that starts with a clearer exploration early in the search process and gradually begins to focus on exploitation towards the end of the search process. This work uses the elites of each knowledge source that are at a certain distance from each other. The dynamic α-cognition procedure assists in providing effective learning of individuals through preserving the diversity of the population during the evolution process. In this procedure, each individual is able to learn from the top α% individuals controlled by its knowledge source in the belief space, where the proportion of the affecting subpopulation (α) is adaptively modified during the evolution. The performance of the proposed work has been evaluated on the CEC’2010 and CEC’2013 benchmark suites developed for the special sessions on large-scale global optimization problems. An appropriate comparative study with the best results in the literature is presented. The results confirm how the merits of the improved Cultural Algorithm can achieve superior performance over other cutting-edge algorithms for these data sets.

Dual Differential Grouping: A More General Decomposition Method for Large-Scale Optimization.

Cooperative coevolution (CC) algorithms based on variable decomposition methods are efficient in solving large-scale optimization problems (LSOPs). However, many decomposition methods, such as the differential grouping (DG) method and its variants, are based on the theorem of function additively separable, which may not work well on problems that are not additively separable and will result in a bottleneck for CC to solve various LSOPs. This deficiency motivates us to study how the decomposition method can decompose more kinds of separable functions, such as the multiplicatively separable function, to improve the general problem-solving ability of CC on LSOPs. With this concern, this article makes the first attempt to decompose multiplicatively separable functions and proposes a novel method called dual DG (DDG) for better LSOP decomposition and optimization. The novelty and advantage of DDG are that it can be suitable for not only additively separable functions but also multiplicatively separable functions, which can considerably expand the application scope of CC. In this article, we will first define the multiplicatively separable function, and then mathematically show its relationship to the additively separable function and how they can be transformed into each other. Based on this, the DDG can use two kinds of differences to detect the separable structure of both additively and multiplicatively separable functions. In addition, the time complexity of DDG is analyzed and a DDG-based CC algorithm framework is developed for solving LSOPs. To verify the superiority of DDG, experiments and comparisons with some state-of-the-art and champion algorithms are conducted not only on 30 LSOPs based on the test suite of the IEEE CEC large-scale global optimization competition, but also on a case study of the parameter optimization for a neural network-based application.

Hybrid Evolutionary Algorithm for Solving the Large-Scale Global Optimization Problems

Hybrid Evolutionary Algorithm for Solving the Large-Scale Global Optimization Problems

A large-scale continuous optimization benchmark suite with versatile coupled heterogeneous modules

The complexity of the engineering design optimizations mainly sources from the large-scale nature of the problems, and the ever-increasing large-scale global optimization (LSGO) problems place high demands on the performance of evolutionary algorithms (EAs). To compare and analyze large-scale optimization algorithms, benchmarks that simulate the features from real-world large-scale optimization are desired. In this paper, we propose a novel large-scale continuous optimization benchmark suite, which includes 15 test functions with the modular structure. The proposed benchmark suite introduces two new features abstracted from the engineering design problems: (1) Heterogeneous design, i.e., the modules are significantly different in terms of difficulty and explicit expressions. (2) Versatile coupling, i.e., different coupling degrees and random coupling topologies of the benchmark are emphasized. Two typical cooperative coevolution frameworks and several state-of-the-art large-scale optimization algorithms are used for comparative studies on the proposed benchmark, and experimental results demonstrate that the proposed benchmark suite is very challenging.

Semi-online parameter identification methodology for maritime power lithium batteries

Due to the special working environment of power lithium battery (P-LiB) on all-electric ships, the high-efficiency battery management system (BMS) is required. In this study, a novel semi-online parameter identification methodology integrated with large-scale global optimization algorithm is developed, in order to ensure the high-quality performance of subsequent BMS functions like the equalization control and state-of-charge estimation. Firstly, the P-LiB parameter identification model is established based on the first-order Thevenin equivalent circuit. Then, the evolution strategy of identification model is developed for dynamically updating the model along with the entire charging/discharging process of P-LiB. Considering that the model complexity increases exponentially with dimensionality, the AMCCDE algorithm developed in our previous work is employed to optimize the dynamic model repeatedly. Moreover, the semi-online operation mechanism for AMCCDE is proposed, in which the multiple context vectors are used to exchange information and inherit the optimal solution between each two adjacent semi-online cycles, and the identification solutions can be dynamically corrected and output at the end of each cycle. Finally, the developed semi-online identification methodology is verified using the USTC-DST and USTC-UDDS datasets. Experimental results show that the developed methodology can well balance the identification accuracy and timeliness, and dynamically output the accurate identification solutions in real-time.

A large-scale global optimization algorithm with a new adaptive computing resource allocation mechanism

A large-scale global optimization algorithm with a new adaptive computing resource allocation mechanism