Merged Differential Grouping for Large-Scale Global Optimization

Abstract

The divide-and-conquer strategy has been widely used in cooperative co-evolutionary algorithms to deal with large-scale global optimization problems, where a target problem is decomposed into a set of lower-dimensional and tractable subproblems to reduce the problem complexity. However, such a strategy usually demands a large number of function evaluations to obtain an accurate variable grouping. To address this issue, a merged differential grouping (MDG) method is proposed in this article based on the subset–subset interaction and binary search. In the proposed method, each variable is first identified as either a separable variable or a nonseparable variable. Afterward, all separable variables are put into the same subset, and the nonseparable variables are divided into multiple subsets using a binary-tree-based iterative merging method. With the proposed algorithm, the computational complexity of interaction detection is reduced to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\max \{n,n_{ns}\times \log _{2} k\})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{ns}(\leq n)$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k( &lt; n)$ </tex-math></inline-formula> indicate the numbers of decision variables, nonseparable variables, and subsets of nonseparable variables, respectively. The experimental results on benchmark problems show that MDG is very competitive with the other state-of-the-art methods in terms of efficiency and accuracy of problem decomposition.