Solving nonlinear optimization problems by Differential Evolution with a rotation-invariant crossover operation using Gram-Schmidt process

Differential Evolution (DE) is a newly proposed evolutionary algorithm. DE is a stochastic direct search method using a population or multiple search points. DE has been successfully applied to optimization problems including nonlinear, non-differentiable, non-convex and multimodal functions. However, the performance of DE degrades in problems with strong linkage among variables, where variables are related strongly each other. One of the desirable properties of optimization algorithms for solving the problems with strong linkage is rotation-invariant property. The rotation-invariant algorithms can solve rotated problems where variables are strongly related as in the same way of solving non-rotated problems. In DE, two operations are applied to each individual: a mutation operation, which is rotation-invariant, and a crossover operation, which is not rotation-invariant. Thus, DE is not rotation-invariant. In this study, we propose a new crossover operation that is rotation-invariant. In order to achieve rotation-invariant property, instead of using the fixed coordinate system, a new coordinate system is build from a current population, or search points in search process. Independent points, or vectors are selected from the population, Gram-Schmidt process is applied to them in order to obtain orthogonal vectors, and the vectors form the new coordinate system. The effect of the rotation-invariant crossover operation is shown by solving some benchmark problems.