Effective decomposition of large-scale separable continuous functions for cooperative co-evolutionary algorithms

Abstract

Abstract-In this paper we investigate the performance of cooperative co-evolutionary (CC) algorithms on large-scale fullyseparable 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.

Designing benchmark problems for large-scale continuous optimization

“…One may assume that the best decomposition for a fully separable problem would be to place each of the decision variables in a separate subcomponent; in other words, decomposing an n-dimensional problem into n 1-dimensional problems. However, this is not necessarily the optimal decomposition [42]. In a cooperative co-evolutionary context, the more subcomponents there are, the more fitness evaluations are needed in one co-evolutionary cycle.…”

“…On the other hand, if all the variables are placed in one large subcomponent, the large size of the search space may render the optimizer inefficient. Some empirical studies suggest that a decomposition between these two extreme cases is perhaps the most efficient [42,61,53]. However, it should be noted that there are a very large number of decomposition possibilities in between these two extremes.…”

“…For a fully separable problem the question of how to decompose it into several smaller subproblems, and the relationship between the dimensionality of each subproblem and the subpopulation size have not been fully answered in the literature [42]. As mentioned previously, when dealing with a non-separable problem, the dimensionality is imposed on the algorithm, and the problem cannot be subdivided any further.…”

“…This essentially reduces these algorithms to a variant of SaNSDE that is applied to a 1000-dimensional problem. It has been shown recently that a better decomposition of separable variables can have a significant effect on the overall optimization performance [42]. MOS that uses a local search has the best performance on this class of problems.…”

“…It is possible to devise a decomposition which do not reflect the actual structure of a benchmark problem, and yet results in a better optimization performance. This happens when there are weakly interacting variables, or when the fully separable variables are subdivided into smaller groups [42].…”