This is an erratum for the article “Introduction to the Special Issue on Large-Scale Optimization and Learning” published in ACM Trans. Evol. Learn. Optim. 5, 4, Article 23 (December 2025), 3 pages.

Introduction to the Special Issue on Large-Scale Optimization and LearningMany real-world optimization problems involve a large number of decision variables.The proliferation of big-data analytic applications has led to the emergence of Large-Scale Optimization Problems (LSOP) at the heart of many data analytics and learning problems [Bottou et al., 2018;Zhou et al., 2014].The curse of dimensionality has made large-scale optimization an exceedingly difficult task, and current optimization methods are often ill-equipped to deal with such problems.To overcome this scalability issue, a wide range of mathematical, metaheuristic, and learningbased optimization algorithms have been developed [Bengio et al., 2021;Omidvar et al., 2022].For example, decomposition methods based on variable interaction analysis have been developed for black-box LSOPs that learn and exploit problem structures [Omidvar et al., 2014].Similarly, there is an emerging set of techniques that leverage Machine Learning (ML) and data mining to significantly reduce the problem size for large-scale combinatorial optimization problems [Sun et al., 2021].This sort of ML-based methods can be incorporated into an optimization algorithm to boost its performance for solving large-scale combinatorial optimization problems [Sun et al., 2022].In recent years, there has been a growing recognition of the synergy between optimization and learning.ML techniques can be used effectively to solve large-scale continuous and combinatorial problems by learning and exploiting problem structures, predicting optimal solutions using data extracted from solved problem instances, reducing problem size, and boosting the performance of existing optimization algorithms.In turn, large-scale optimization can help ML by providing the backbone for many learning algorithms and applications.For example, large-scale optimization is essential for training deep neural networks, where optimization problems with billions of decision variables need to be solved [Hinton and Salakhutdinov, 2006].Therefore, the integration of optimization and learning will play a critical role in addressing the scalability issues that arise in many real-world applications.This special issue was conceived to fill this gap in research that explores the intersection between these two fields.After a rigorous peer-review process, it is our greatest pleasure to introduce the five accepted papers in this special issue.In "Island-Based Evolutionary Computation with Diverse Surrogates and Adaptive Knowledge Transfer for High-Dimensional Data-Driven Optimization" by Xian-Rong Zhang, Yue-Jiao Gong, Zhiguang Cao, and Jun Zhang, an offline data-driven evolutionary algorithm is proposed to make use of surrogate models to approximate an objective function with only a limited amount of

Clustering in dynamic environments is of increasing importance, with broad applications ranging from real-time data analysis and online unsupervised learning to dynamic facility location problems. While meta-heuristics have shown promising effectiveness in static clustering tasks, their application for tracking optimal clustering solutions or robust clustering over time in dynamic environments remains largely underexplored. This is partly due to a lack of dynamic datasets with diverse, controllable, and realistic dynamic characteristics, hindering systematic performance evaluations of clustering algorithms in various dynamic scenarios. This deficiency leads to a gap in our understanding and capability to effectively design algorithms for clustering in dynamic environments. To bridge this gap, this paper introduces the Dynamic Dataset Generator (DDG). DDG features multiple dynamic Gaussian components integrated with a range of heterogeneous, local, and global changes. These changes vary in spatial and temporal severity, patterns, and domain of influence, providing a comprehensive tool for simulating a wide range of dynamic scenarios.

Robust optimization over time (ROOT) is the combination of robust optimization and dynamic optimization. In ROOT, frequent changes to deployed solutions are undesirable, which can be due to the high cost of switching between deployed solutions, limitations on the resources required to deploy new solutions, and/or the system’s inability to tolerate frequent changes in the deployed solutions. ROOT is dedicated to the study and development of algorithms capable of dealing with the implications of deploying or maintaining solutions over longer time horizons involving multiple environmental changes. This paper presents an in-depth review of the research on ROOT. The overarching aim of this survey is to help researchers gain a broad perspective on the current state of the field, what has been achieved so far, and the existing challenges and pitfalls. This survey also aims to improve accessibility and clarity by standardizing terminology and unifying mathematical notions used across the field, providing explicit mathematical formulations of definitions, and improving many existing mathematical descriptions. Moreover, we classify ROOT problems based on two ROOT-specific criteria: the requirements for changing or keeping deployed solutions and the number of deployed solutions. This classification helps researchers gain a better understanding of the characteristics and requirements of ROOT problems, which is crucial to systematic algorithm design and benchmarking. Additionally, we classify ROOT methods based on the approach they use for finding robust solutions and provide a comprehensive review of them. This survey also reviews ROOT benchmarks and performance indicators. Finally, we identify several future research directions.

Many real-world optimization problems are dynamic. The field of robust optimization over time (ROOT) deals with dynamic optimization problems in which frequent changes of the deployed solution are undesirable. This can be due to the high cost of switching the deployed solutions, the limitation of the needed resources to deploy such new solutions, and/or the system being intolerant towards frequent changes of the deployed solution. In the considered ROOT problems in this article, the main goal is to find solutions that maximize the average number of environments where they remain acceptable. In the state-of-the-art methods developed to tackle these problems, the decision makers/metrics used to select solutions for deployment mostly make simplifying assumptions about the problem instances. Besides, the current methods all use the population control components which have been originally designed for tracking the global optimum over time without taking any robustness considerations into account. In this paper, a multi-population ROOT method is proposed with two novel components: a robustness estimation component that estimates robustness of the promising regions, and a dual-mode computational resource allocation component to manage sub-populations by taking several factors, including robustness, into account. Our experimental results demonstrate the superiority of the proposed method over other state-of-the-art approaches.

Population clustering methods, which consider the position and fitness of individuals to form sub-populations in multi-population algorithms, have shown high efficiency in tracking the moving global optimum in dynamic optimization problems. However, most of these methods use a fixed population size, making them inflexible and inefficient when the number of promising regions is unknown. The lack of a functional relationship between the population size and the number of promising regions significantly degrades performance and limits an algorithm’s agility to respond to dynamic changes. To address this issue, we propose a new species-based particle swarm optimization with adaptive population size and number of sub-populations for solving dynamic optimization problems. The proposed algorithm also benefits from a novel systematic adaptive deactivation component that, unlike the previous deactivation components, adapts the computational resource allocation to the sub-populations by considering various characteristics of both the problem and the sub-populations. We evaluate the performance of our proposed algorithm for the Generalized Moving Peaks Benchmark and compare the results with several peer approaches. The results indicate the superiority of the proposed method.

Dynamic changes are an important and inescapable aspect of many real-world optimization problems. Designing algorithms to find and track desirable solutions while facing challenges of dynamic optimization problems is an active research topic in the field of swarm and evolutionary computation. To evaluate and compare the performance of algorithms, it is imperative to use a suitable benchmark that generates problem instances with different controllable characteristics. In this article, we give a comprehensive review of existing benchmarks and investigate their shortcomings in capturing different problem features. We then propose a highly configurable benchmark suite, the generalized moving peaks benchmark, capable of generating problem instances whose components have a variety of properties, such as different levels of ill-conditioning, variable interactions, shape, and complexity. Moreover, components generated by the proposed benchmark can be highly dynamic with respect to the gradients, heights, optimum locations, condition numbers, shapes, complexities, and variable interactions. Finally, several well-known optimizers and dynamic optimization algorithms are chosen to solve generated problems by the proposed benchmark. The experimental results show the poor performance of the existing methods in facing new challenges posed by the addition of new properties.

Scalability of optimization algorithms is a major challenge in coping with the ever-growing size of optimization problems in a wide range of application areas from high-dimensional machine learning to complex large-scale engineering problems. The field of large-scale global optimization is concerned with improving the scalability of global optimization algorithms, particularly, population-based metaheuristics. Such metaheuristics have been successfully applied to continuous, discrete, or combinatorial problems ranging from several thousand dimensions to billions of decision variables. In this two-part survey, we review recent studies in the field of large-scale black-box global optimization to help researchers and practitioners gain a bird’s-eye view of the field, learn about its major trends, and the state-of-the-art algorithms. Part I of the series covers two major algorithmic approaches to large-scale global optimization: 1) problem decomposition and 2) memetic algorithms. Part II of the series covers a range of other algorithmic approaches to large-scale global optimization, describes a wide range of problem areas, and finally, touches upon the pitfalls and challenges of current research and identifies several potential areas for future research.

This article is the second part of a two-part survey series on large-scale global optimization. The first part covered two major algorithmic approaches to large-scale optimization, namely, decomposition methods and hybridization methods, such as memetic algorithms and local search. In this part, we focus on sampling and variation operators, approximation and surrogate modeling, initialization methods, and parallelization. We also cover a range of problem areas in relation to large-scale global optimization, such as multiobjective optimization, constraint handling, overlapping components, the component imbalance issue and benchmarks, and applications. The article also includes a discussion on pitfalls and challenges of the current research and identifies several potential areas of future research.

Scalability is a crucial aspect of designing efficient algorithms. Despite their prevalence, large-scale dynamic optimization problems are not well studied in the literature. This paper is concerned with designing benchmarks and frameworks for the study of large-scale dynamic optimization problems. We start by a formal analysis of the moving peaks benchmark (MPB) and show its nonseparable nature irrespective of its number of peaks. We then propose a composite MPB suite with exploitable modularity covering a wide range of scalable partially separable functions suitable for the study of large-scale dynamic optimization problems. The benchmark exhibits modularity, heterogeneity, and imbalance features to resemble real-world problems. To deal with the intricacies of large-scale dynamic optimization problems, we propose a decomposition-based coevolutionary framework which breaks a large-scale dynamic optimization problem into a set of lower-dimensional components. A novel aspect of the framework is its efficient bi-level resource allocation mechanism which controls the budget assignment to components and the populations responsible for tracking multiple moving optima. Based on a comprehensive empirical study on a wide range of large-scale dynamic optimization problems with up to 200-D, we show the crucial role of problem decomposition and resource allocation in dealing with these problems. The experimental results clearly show the superiority of the proposed framework over three other approaches in solving large-scale dynamic optimization problems.

In this paper we use a divide-and-conquer approach to tackle large-scale optimization problems with overlapping components. Decomposition for an overlapping problem is challenging as its components depend on one another. The existing decomposition methods typically assign all the linked decision variables into one group, thus cannot reduce the original problem size. To address this issue we modify the Recursive Differential Grouping (RDG) method to decompose overlapping problems, by breaking the linkage at variables shared by multiple components. To evaluate the efficacy of our method, we extend two existing overlapping benchmark problems considering various level of overlap. Experimental results show that our method can greatly improve the search ability of an optimization algorithm via divide-and-conquer, and outperforms RDG, random decomposition as well as other state-of-the-art methods. We further evaluate our method using the CEC’2013 benchmark problems and show that our method is very competitive when equipped with a component optimizer.

Engineering designs can involve multiple stages, where at each stage, the design models are incrementally modified and optimized. In contrast to traditional dynamic optimization problems, where the changes are caused by some objective factors, the changes in such incremental optimization problems (IOPs) are usually caused by the modifications made by the decision makers during the design process. While existing work in the literature is mainly focused on traditional dynamic optimization, little research has been dedicated to solving such IOPs. In this paper, we study how to adopt cooperative coevolution to efficiently solve a specific type of IOPs, namely, those with increasing decision variables. First, we present a benchmark function generator on the basis of some basic formulations of IOPs with increasing decision variables and exploitable modular structure. Then, we propose a contribution-based cooperative coevolutionary framework coupled with an incremental grouping method for dealing with them. On one hand, the benchmark function generator is capable of generating various benchmark functions with various characteristics. On the other hand, the proposed framework is promising in solving such problems in terms of both optimization accuracy and computational efficiency. In addition, the proposed method is further assessed using a real-world application, i.e., the design optimization of a stepped cantilever beam.

Problem decomposition plays an essential role in the success of cooperative co-evolution (CC), when used for solving large-scale optimization problems. The recently proposed recursive differential grouping (RDG) method has been shown to be very efficient, especially in terms of time complexity. However, it requires an appropriate parameter setting to estimate a threshold value in order to determine if two subsets of decision variables interact or not. Furthermore, using one global threshold value may be insufficient to identify variable interactions in components with different contribution to the fitness value. Inspired by the different grouping 2 (DG2) method, in this paper, we adaptively estimates a threshold value based on computational round-off errors for RDG. We derive an upper bound of the round-off errors, which is shown to be sufficient when identifying variable interactions across a wide range of large-scale benchmark problems. Comprehensive numerical experimental results showed that the proposed RDG2 method achieved higher decomposition accuracy than RDG and DG2. When embedded into a CC framework, it achieved statistically equal or significantly better solution quality than RDG and DG2, when used to solve the benchmark problems.

Dynamic optimization problems (DOPs) are problems that change over time. However, most investigations in this domain are focused on tracking moving optima (TMO) without considering the cost of switching from one solution to another when the environment changes. Robust optimization over time (ROOT) tries to address this shortcoming by finding solutions which remain acceptable for several environments. However, ROOT methods change solutions only when they become unacceptable. Indeed, TMO and ROOT are two extreme cases in the sense that in the former, the switching cost is considered zero and in the latter, it is considered very large. In this paper, we propose a new semi ROOT algorithm based on a new approach to switching cost. This algorithm changes solutions when: 1) the current solution is not acceptable and 2) the current solution is still acceptable but algorithm has found a better solution and switching is preferable despite the cost. The main objective of the proposed algorithm is to maximize the performance based on the fitness of solutions and their switching cost. The experiments are done on modified moving peaks benchmark (mMPB) and the performance of the proposed algorithm alongside state-of-the-art ROOT and TMO methods is investigated.

A proportional-integral-derivative (PID) controller is a control loop feedback mechanism widely employed in industrial control systems. The parameters tuning is a sticking point, having a great effect on the control performance of a PID system. There is no perfect rule for designing controllers, and finding an initial good guess for the parameters of a well-performing controller is difficult. In this paper, we develop a knowledge-based particle swarm optimization by incorporating the dynamic response information of PID into the optimizer. Prior knowledge not only empowers the particle swarm optimization algorithm to quickly identify the promising regions, but also helps the proposed algorithm to increase the solution precision in the limited running time. To benchmark the performance of the proposed algorithm, an electric pump drive and an automatic voltage regulator system are selected from industrial applications. The simulation results indicate that the proposed algorithm with a newly proposed performance index has a significant performance on both test cases and outperforms other algorithms in terms of overshoot, steady state error, and settling time.

Identification of variable interaction is essential for an efficient implementation of a divide-and-conquer algorithm for large-scale black-box optimization. In this paper, we propose an improved variant of the differential grouping (DG) algorithm, which has a better efficiency and grouping accuracy. The proposed algorithm, DG2, finds a reliable threshold value by estimating the magnitude of roundoff errors. With respect to efficiency, DG2 reuses the sample points that are generated for detecting interactions and saves up to half of the computational resources on fully separable functions. We mathematically show that the new sampling technique achieves the lower bound with respect to the number of function evaluations. Unlike its predecessor, DG2 checks all possible pairs of variables for interactions and has the capacity to identify overlapping components of an objective function. On the accuracy aspect, DG2 outperforms the state-of-the-art decomposition methods on the latest large-scale continuous optimization benchmark suites. DG2 also performs reliably in the presence of imbalance among contribution of components in an objective function. Another major advantage of DG2 is the automatic calculation of its threshold parameter ($\epsilon $ ), which makes it parameter-free. Finally, the experimental results show that when DG2 is used within a cooperative co-evolutionary framework, it can generate competitive results as compared to several state-of-the-art algorithms.

Cooperative co-evolution (CC) is an explicit means of problem decomposition in multipopulation evolutionary algorithms for solving large-scale optimization problems. For CC, subpopulations representing subcomponents of a large-scale optimization problem co-evolve, and are likely to have different contributions to the improvement of the best overall solution to the problem. Hence, it makes sense that more computational resources should be allocated to the subpopulations with greater contributions. In this paper, we study how to allocate computational resources in this context and subsequently propose a new CC framework named CCFR to efficiently allocate computational resources among the subpopulations according to their dynamic contributions to the improvement of the objective value of the best overall solution. Our experimental results suggest that CCFR can make efficient use of computational resources and is a highly competitive CCFR for solving large-scale optimization problems.

Competition in cooperative coevolution (CC) has demonstrated success in solving global optimization problems. In a recent study, a multi-island competitive cooperative coevolution (MIC3) algorithm was introduced that featured competition and collaboration of several different problem decomposition strategies implemented as independent islands. It was shown that MIC3converges to high quality solutions without the need to find an optimal decomposition. MIC3splits the computational budget in terms of the number of function evaluations, equally amongst all the islands and evolves them in a round-robin fashion. This overlooks the difference in contributions of different islands towards improving the overall objective function value. Therefore, a considerable amount of function evaluations is wasted on the low-contributing islands as their problem decomposition strategies may not appeal to the problem at the given stage of the evolutionary process. This paper proposes contribution-based MIC3algorithms (MIC4) that quantifies the contributions of each island and allocates the computational budget accordingly. The experimental analysis reveals that the proposed method outperforms its counterpart.

Cooperative Co-evolution (CC) is a promising framework for solving large-scale optimization problems. However, the round-robin strategy of CC is not an efficient way of allocating the available computational resources to components of imbalanced functions. The imbalance problem happens when the components of a partially separable function have non-uniform contributions to the overall objective value. Contribution-Based Cooperative Co-evolution (CBCC) is a variant of CC that allocates the available computational resources to the individual components based on their contributions. CBCC variants (CBCC1 and CBCC2) have shown better performance than the standard CC in a variety of cases. In this paper, we show that over-exploration and over-exploitation are two major sources of performance loss in the existing CBCC variants. On that basis, we propose a new contribution-based algorithm that maintains a better balance between exploration and exploitation. The empirical results show that the new algorithm is superior to its predecessors as well as the standard CC.

This article proposes a competitive divide-and-conquer algorithm for solving large-scale black-box optimization problems for which there are thousands of decision variables and the algebraic models of the problems are unavailable. We focus on problems that are partially additively separable, since this type of problem can be further decomposed into a number of smaller independent subproblems. The proposed algorithm addresses two important issues in solving large-scale black-box optimization: (1) the identification of the independent subproblems without explicitly knowing the formula of the objective function and (2) the optimization of the identified black-box subproblems. First, a Global Differential Grouping (GDG) method is proposed to identify the independent subproblems. Then, a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is adopted to solve the subproblems resulting from its rotation invariance property. GDG and CMA-ES work together under the cooperative co-evolution framework. The resultant algorithm, named CC-GDG-CMAES, is then evaluated on the CEC’2010 large-scale global optimization (LSGO) benchmark functions, which have a thousand decision variables and black-box objective functions. The experimental results show that, on most test functions evaluated in this study, GDG manages to obtain an ideal partition of the index set of the decision variables, and CC-GDG-CMAES outperforms the state-of-the-art results. Moreover, the competitive performance of the well-known CMA-ES is extended from low-dimensional to high-dimensional black-box problems.

Cooperative Co-evolutionary (CC) techniques have demonstrated the promising performance in dealing with large-scale optimization problems. However, in many applications, their performance may drop due to the presence of imbalanced contributions to the objective function value from different subsets of decision variables. To remedy this drawback, Contribution-Based Cooperative Co-evolutionary (CBCC) algorithms have been proposed. They have presented significant improvements over traditional CC techniques when the decomposition is accurate and the imbalance level is very high. However, in real-world scenarios, we might not have the knowledge about the ideal decomposition and actual imbalance level of a problem to be solved. Therefore, this study aims at analysing the performance of existing CBCC techniques in more realistic settings, i.e., when the decomposition error is unavoidable and the imbalance level is low or moderate. Our in-depth analysis reveals that even in these situations, CBCC algorithms are superior alternatives to traditional CC techniques. We also observe that the variations of CBCC techniques may lead to the significantly different performance. Thus, we recommend practitioners to carefully choose a competent variant of CBCC which best suits their particular applications.

MOEA/D is an evolutionary multi-objective optimization algorithm, which relies on decomposition methods such as, weighted-sum, Tchebycheff and Penalty-based Boundary Intersection (PBI) to convert a multi-objective problem into a set of single-objective problems. It is known that PBI can generate a more uniform set of solutions than the other decomposition methods. The drawback of PBI is that it has a penalty parameter (θ) that has to be specified by the user. This penalty parameter can affect the convergence rate of MOEA/D as well as the uniformity of solutions. Unfortunately, there are very limited studies on sensitivity analysis of MOEA/D on the penalty parameter of PBI. This paper is dedicated to a comprehensive analysis of PBI's penalty parameter, and its effect on a user-preference algorithm (R-MEAD2) and a non-user-preference algorithm (MOEA/D). Unlike the previous studies that only rely on Hypervolume as their performance measure, we study the effect of θ on convergence, uniformity, and the combination of convergence and uniformity independently. The experimental results suggest that user-preference algorithms consistently perform better with a relatively larger θ value as compared to their non-user-preference counterparts. The results also suggest that on some problems, such as multi-modal functions, convergence is the dominant factor on the overall performance, where a smaller θ is preferable. Conversely, on some other problems, a larger θ is suggested where uniformity is the dominant factor. Finally, we briefly investigate the relationship between θ and the number of objectives.

Opposition-based learning (OBL) and cooperative co-evolution (CC) have demonstrated promising performance when dealing with large-scale global optimization (LSGO) problems. In this work, we propose a novel framework for hybridizing these two techniques, and investigate the performance of simple implementations of this new framework using the most recent LSGO benchmarking test suite. The obtained results verify the effectiveness of our proposed OBL-CC framework. Moreover, some advanced statistical analyses reveal that the proposed hybridization significantly outperforms its component methods in terms of the quality of finally obtained solutions.

Evolutionary algorithms that rely on dominance ranking often suffer from a low selection pressure problem when dealing with many-objective problems. Decomposition and user-preference based methods can help to alleviate this problem to a great extent. In this paper, a user-preference based evolutionary multi-objective algorithm is proposed that uses decomposition methods for solving many-objective problems. Decomposition techniques that are widely used in multi-objective evolutionary optimization require a set of evenly distributed weight vectors to generate a diverse set of solutions on the Pareto-optimal front. The newly proposed algorithm, R-MEAD2, improves the scalability of its previous version, R-MEAD, which uses a simplexlattice design method for generating weight vectors. This makes the population size is dependent on the dimension size of the objective space. R-MEAD2 uses a uniform random number generator to remove the coupling between dimension and the population size. This paper shows that a uniform random number generator is simple and able to generate evenly distributed points in a high dimensional space. Our comparative study shows that R-MEAD2 outperforms the dominance-based method R-NSGA-II on many-objective problems.

In this paper we investigate the performance of cooperative co-evolutionary (CC) algorithms on large-scale fully-separable continuous optimization problems. We have shown that decomposition can have significant impact on the performance of CC algorithms. The empirical results show that the subcomponent size should be chosen small enough so that the subcomponent size is within the capacity of the subcomponent optimizer. In practice, determining the optimal size is difficult. Therefore, adaptive techniques are desired by practitioners. Here we propose an adaptive method, MLSoft, that uses widely-used techniques in reinforcement learning such as the value function method and softmax selection rule to adapt the subcomponent size during the optimization process. The experimental results show that MLSoft is significantly better than an existing adaptive algorithm called MLCC on a set of large-scale fully-separable problems.

Cooperative co-evolution has been introduced into evolutionary algorithms with the aim of solving increasingly complex optimization problems through a divide-and-conquer paradigm. In theory, the idea of co-adapted subcomponents is desirable for solving large-scale optimization problems. However, in practice, without prior knowledge about the problem, it is not clear how the problem should be decomposed. In this paper, we propose an automatic decomposition strategy called differential grouping that can uncover the underlying interaction structure of the decision variables and form subcomponents such that the interdependence between them is kept to a minimum. We show mathematically how such a decomposition strategy can be derived from a definition of partial separability. The empirical studies show that such near-optimal decomposition can greatly improve the solution quality on large-scale global optimization problems. Finally, we show how such an automated decomposition allows for a better approximation of the contribution of various subcomponents, leading to a more efficient assignment of the computational budget to various subcomponents.

In this paper, we propose a metric for evaluating the performance of user-preference based evolutionary multiobjective algorithms by defining a preferred region based on the location of a user-supplied reference point. This metric uses a composite front which is a type of reference set and is used as a replacement for the Pareto-optimal front. This composite front is constructed by extracting the non-dominated solutions from the merged solution sets of all algorithms that are to be compared. A preferred region is then defined on the composite front based on the location of a reference point. Once the preferred region is defined, existing evolutionary multi-objective performance metrics can be applied with respect to the preferred region. In this paper the performance of a cardinality-based metric, a distance-based metric, and a volume-based metric are compared against a baseline which relies on knowledge of the Pareto-optimal front. The experimental results show that the distance-based and the volume-based metrics are consistent with the baseline, showing meaningful comparisons. However, the cardinality-based approach shows some inconsistencies and is not suitable for comparing the algorithms.

In this paper we propose a user-preference based evolutionary algorithm that relies on decomposition strategies to convert a multi-objective problem into a set of single-objective problems. The use of a reference point allows the algorithm to focus the search on more preferred regions which can potentially save considerable amount of computational resources. The algorithm that we proposed, dynamically adapts the weight vectors and is able to converge close to the preferred regions. Combining decomposition strategies with reference point approaches paves the way for more effective optimization of many-objective problems. The use of a decomposition method alleviates the selection pressure problem associated with dominance-based approaches while a reference point allows a more focused search. The experimental results show that the proposed algorithm is capable of finding solutions close to the reference points specified by a decision maker. Moreover, our results show that high quality solutions can be obtained using less computational effort as compared to a state-of-the-art decomposition based evolutionary multi-objective algorithm.

Standard Cooperative Co-evolution uses a round-robin method to select subcomponents to undergo optimization. In a non-separable (epistatic) optimization problem, dividing the computational budget equally between all of the subcomponents is not necessarily the best strategy. When dealing with non-separable problems, there is usually an imbalance between the contribution of various subcomponents to the global fitness of the individuals. Using a round-robin fashion treats all of the subcomponents equally and wastes the computational budget. In this paper, we propose a Contribution Based Cooperative Co-evolution (CBCC) that selects the subcomponents based on their contributions to the global fitness. This alleviates the imbalance issue and allows the computational resources to be used more efficiently. Experiments on several benchmark functions with the "imbalance issue" show that this new scheme is promising, especially when it is combined with a grouping algorithm that captures interacting variables in common subcomponents.

Many evolutionary algorithms have been proposed for large scale optimization. Parameter interaction in non-separable problems is a major source of performance loss specially on large scale problems. Cooperative Co-evolution(CC) has been proposed as a natural solution for large scale optimization problems, but lack of a systematic way of decomposing large scale non-separable problems is a major obstacle for CC frameworks. The aim of this paper is to propose a systematic way of capturing interacting variables for a more effective problem decomposition suitable for cooperative co-evolutionary frameworks. Grouping interacting variables in different subcomponents in a CC framework imposes a limit to the extent interacting variables can be optimized to their optimum values, in other words it limits the improvement interval of interacting variables. This is the central idea of the newly proposed technique which is called delta method. Delta method measures the averaged difference in a certain variable across the entire population and uses it for identifying interacting variables. The experimental results show that this new technique is more effective than the existing random grouping method.

In this paper we propose three techniques to improve the performance of one of the major algorithms for large scale continuous global function optimization. Multilevel Cooperative Co-evolution (MLCC) is based on a Cooperative Co-evolutionary framework and employs a technique called random grouping in order to group interacting variables in one subcomponent. It also uses another technique called adaptive weighting for co-adaptation of subcomponents. We prove that the probability of grouping interacting variables in one subcomponent using random grouping drops significantly as the number of interacting variables increases. This calls for more frequent random grouping of variables. We show how to increase the frequency of random grouping without increasing the number of fitness evaluations. We also show that adaptive weighting is ineffective and in most cases fails to improve the quality of found solution, and hence wastes considerable amount of CPU time by extra evaluations of objective function. Finally we propose a new technique for self-adaptation of the subcomponent sizes in CC. We demonstrate how a substantial improvement can be gained by applying these three techniques.