CCMBO: a covariance-based clustered monarch butterfly algorithm for optimization problems

Rotationally variance nature-inspired algorithms are not efficient for solving non-separable problems. One way for solving this limitation is utilizing the concept of covariance-based learning to transform the original space into the new space in which the interactions among variables are revealed and operators perform in an appropriate coordinate system. In this paper, Monarch butterfly optimization (MBO), a new nature-inspired and rotation-variance algorithm, is studied. By focusing on making MBO more rotationally invariant, a covariance-based clustered MBO (CCMBO) is presented. In the CCMBO, two primary operators of MBO are modified. An eigenvector-based migration operator and a linearized adjusting operator are utilized to make MBO more rotationally invariant. CCMBO employs a re-initialization operator to improve its exploration ability. Also, to allow exploiting obtained information about the search space, CCMBO utilizes self-organizing map clustering. The CCMBO is evaluated on an extensive set of optimization benchmark functions. It is compared with MBO, two of its improvements, and six other state-of-the-art evolutionary algorithms. The results illustrate that CCMBO obtains significantly better performance and would be a valuable and practical algorithm for optimization problems.