Function Optimization in Conformal Space by Using Spherical Inversions and Reflections

Abstract

This paper introduces an Evolutionary Algorithm in Conformal Space (EACS) for global continuous optimization and its implementation by using Conformal Geometric Algebra (CGA). Two new geometric search operators are included in the design of the EACS: Inversion Search Operator (ISO) and Reflection Search Operator (RSO). The ISO computes the inverse points with respect to hyper-spheres, and the RSO redistributes the individuals on the surface of the hyper-sphere. The nonlinear geometric nature of the ISO furnishes and enhances the search capability of the algorithm. The reproduction operators are described in the framework of the CGA. CGA provides a concise way to perform rigid euclidean transformations(rotations, translations, reflections) and inversions on hyper-spheres. These transformations are easily computed by using the products of the CGA. The performance of the EACS is analyzed through a benchmark of 28 functions. Statistical tests show the competitive performance of EACS in comparison with current leading algorithms (PSO and DE).