Self-adaptive Based Cooperative Coevolutionary Algorithm for Large-scale Numerical Optimization

Cooperative Coevolution (CC) framework has become a powerful approach to solve large-scale global optimization problems effectively. Although a number of significant modifications of CC algorithms have been introduced in recent years, the theoretical studies of population initialization strategies in the CC framework are quite limited so far. The population initialization strategies can help a population-based algorithm to start with better candidate solutions for achieving better results. In this paper, we propose a CC algorithm with population initialization strategies based on the center region to improve its performance. Three population initialization strategies, namely, center-based normal distribution sampling, central golden region, and hybrid random-center normal distribution sampling are utilized in the CC framework. These population initialization strategies attempt to generate points around center-point with different schemes. The performance of the proposed algorithm is evaluated on CEC-2013 LSGO benchmark functions. Simulation results confirm that the proposed algorithm obtains a promising performance on the majority of the nonseparable high dimension benchmark functions.

This paper presents the Constructive Cooperative Coevolutionary (mathrm {C}^3) algorithm, applied to continuous large-scale global optimisation problems. The novelty of mathrm {C}^3 is that it utilises a multi-start architecture and incorporates the Cooperative Coevolutionary algorithm. The considered optimisation problem is decomposed into subproblems. An embedded optimisation algorithm optimises the subproblems separately while exchanging information to co-adapt the solutions for the subproblems. Further, mathrm {C}^3 includes a novel constructive heuristic that generates different feasible solutions for the entire problem and thereby expedites the search. In this work, two different versions of mathrm {C}^3 are evaluated on high-dimensional benchmark problems, including the CEC’2013 test suite for large-scale global optimisation. mathrm {C}^3 is compared with several state-of-the-art algorithms, which shows that mathrm {C}^3 is among the most competitive algorithms. mathrm {C}^3 outperforms the other algorithms for most partially separable functions and overlapping functions. This shows that mathrm {C}^3 is an effective algorithm for large-scale global optimisation. This paper demonstrates the enhanced performance by using constructive heuristics for generating initial feasible solutions for Cooperative Coevolutionary algorithms in a multi-start framework.

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.

This paper attempts to address the problem of large scale optimization and high dimensional optimization using principal component analysis (PCA) strategy with differential evolution (DE) based on Cooperative Co-evolution (CC) framework. Decomposition problem is a major obstacle for large-scale optimization problems. The aim of this paper is to propose effective dimension decomposition method of PCA strategy for capturing the main information among dimensions. PCA strategy can measures most of the contribution information of dimension and uses it for identifying main dimension to guide them to group the most promising subcomponents in CC framework. Then each subcomponents can be solved using an evolutionary optimizer to find the optimum values. The experimental results show that this new technique is more effective than some existing grouping methods.

Although numbers of heuristic algorithms are successfully developed for solving portfolio optimization problems, this is not for all cases of the large-scale ones. A large-scale portfolio optimization involves dealing with the large search space and dense variance-covariance matrix associated with the problem. This paper proposed a new multi-objective algorithm for solving a large-scale optimization problem based upon the notion of cooperative coevolutionary algorithms (CCA). The new problem decomposition scheme was designed by allowing the species-size to be dynamically adjusted as the runs progress. This scheme enhances capability of traditional CCA in dealing with non-separable optimization problem. The collaborator selection method was modified to allow the proposed CCA to perform in a multi-objective (MO) optimization framework. Additionally, the proposed algorithm, named as “DMOCCA”, was implemented for solving large-scale portfolio optimization problem with cardinality constraint using the real-world data set having scale up to 2196 dimensions. Moreover, its performances were benchmarked with those of the SPEA-II and MOPSO.

Cooperative coevolutionary algorithm with resource allocation strategies to minimize unnecessary computations

Conflict resolution becomes a worldwide urgent problem to guarantee the airspace safety. The existing approaches are mostly short-term or middle-term which obtain solutions by local adjustment. 4D-Trajectory conflict resolution (4DTCR), as a long-term method, can give better solutions to all flights in a global view. 4DTCR involved with China air route network and thousands of flight plans is a large and complex problem which is hard to be solved by classical approaches. In this paper, the cooperative coevolution (CC) algorithm with random grouping strategy is presented for its advantage in dealing with large and complex problem. Moreover, a fast Genetic Algorithm (GA) is designed for each subcomponent optimization which is effective and efficient to obtain optimal solution. Experimental studies are conducted to compare it to the genetic algorithm in previous approach and CC algorithm with classic grouping strategy. The results show that our algorithm has a better performance.Keywords4D-TrjactoryConflict resolutionCooperative coevolutionEvolutionary computation

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

The rapid progress of modern technologies generates a massive amount of high-throughput data, called Big Data, which provides opportunities to find new insights using machine learning (ML) algorithms. Big Data consist of many features (also called attributes); however, not all these are necessary or relevant, and they may degrade the performance of ML algorithms. Feature selection (FS) is an essential preprocessing step to reduce the dimensionality of a dataset. Evolutionary algorithms (EAs) are widely used search algorithms for FS. Using classification accuracy as the objective function for FS, EAs, such as the cooperative co-evolutionary algorithm (CCEA), achieve higher accuracy, even with a higher number of features. Feature selection has two purposes: reducing the number of features to decrease computations and improving classification accuracy, which are contradictory but can be achieved using a single objective function. For this very purpose, this paper proposes a penalty-based wrapper objective function. This function can be used to evaluate the FS process using CCEA, hence called Cooperative Co-Evolutionary Algorithm-Based Feature Selection (CCEAFS). An experiment was performed using six widely used classifiers on six different datasets from the UCI ML repository with FS and without FS. The experimental results indicate that the proposed objective function is efficient at reducing the number of features in the final feature subset without significantly reducing classification accuracy. Based on different performance measures, in most cases, naïve Bayes outperforms other classifiers when using CCEAFS.

A local cooperative approach to solve large-scale constrained optimization problems

Large-scale global optimization (LSGO) problems are emergent in many domains of applied sciences. LSGO is a hard challenge for the majority of state-of-the-art optimization methods. Evolution algorithms (EAs) combined with Cooperative Coevolution (CC) are able to perform well when solving many real-world LSGO problems. The grouping of variables at the problem decomposition stage has a significant impact on the performance of the CC approach. This study proposes an improvement of the previously developed random adaptive grouping (RAG) approach for CC. The new method is titled as RAG2, and the whole optimization algorithm, based on RAG2, is called CC-SHADE-RAG2. The influence of the choice of the population size and the number of subcomponents on the algorithm performance have been investigated using the IEEE LSGO CEC’2013 benchmark. The set of test problems in the benchmark contains fifteen functions with dimensionality equal to one thousand. We have also compared the performance of the novel algorithm with some EAs, which are applied for solving LSGO problems.

It has been shown that cooperative coevolution (CC) can effectively deal with large scale optimization problems (LSOPs) through a divide-and-conquer strategy. However, its performance is severely restricted by the current context-vector-based sub-solution evaluation method because this method requires too many computation resources. To alleviate this issue, this study proposes an adaptive surrogate model assisted CC framework which adaptively constructs surrogate models for different sub-problems by fully considering their characteristics. By this means, the computation cost could be greatly reduced without significantly sacrificing evaluation quality. Empirical studies on IEEE CEC 2010 large scale benchmark suit show that the concrete algorithm based on this framework performs well.

The solution of large-scale optimization problems is the key to many decision-making processes in practice. However, it is a challenging research topic when considered both the quality of solutions and the required computational time. One of the popular approaches for these problems is to divide the problems into a number of smaller sub-problems, that are then solved separately with an exchange of some information using the cooperative co-evolution (CC) concept. However, the characteristics of sub-components could be different, and their contributions to the overall performance can also be different while solving the problem. In the CC approach, it usually applies one optimizer and allocates equal computational budget to all sub-components. In this article, a new algorithm is proposed with the use of multiple optimizers, along with a need-based allocation of computational budget for the sub-components. In the proposed algorithm, a group of optimizers cooperate in an effective way to evolve the sub-components, depending on heuristic fuzzy rules. The performance of our proposed algorithm was evaluated by solving a number of large-scale global optimization benchmark functions. The empirical results show that the proposed algorithm outperforms equal allocation CC, a single selection characteristic, a single candidate optimizer and state-of-the-art algorithms.

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.

Coevolutionary algorithms offer great promise as adaptive problem solvers but suffer from several known pathologies. Historically, spatially embedded coevolutionary algorithms seem to have succeeded where other coevolutionary approaches fail; however, explanations for this have been largely unexplored. We examine this idea more closely by looking at spatial models in the context of a particular coevolutionary pathology: loss of gradient. We believe that loss of gradient in cooperative coevolution is caused by asymmetries in the problem or initial conditions between populations, driving one population to convergence before another. Spatial models seem to lock populations together in terms of evolutionary change, helping establish a type of dynamic balance to thwart loss of gradient. We construct a tunably asymmetric function optimization problem domain and conduct an empirical study to justify this assertion. We find that spatial restrictions for collaboration and selection can help keep population changes balanced when presented with severe asymmetries in the problem.