Nonlinear equation systems solved by many-objective Hype

Abstract

A difficulty in solving nonlinear equation systems (NESs) stays in finding all the solutions for NES. This paper uses multi-objective evolutionary techniques to overcome it. We converted the NES into a multi-objective optimization problem (MOP) with a parameter C. The Pareto-optimal set of the MOP becomes the solutions of the NES when the parameter C gets to infinity. Next, a multi-objective evolutionary algorithm (MOEA) is used to solve the transformed MOP, during which C is gradually approaching infinity. A significant feature of this algorithm is that there is one-to-one relationship between the Pareto optimal set and the Pareto front, which suggests that different solutions have different objective values in the MOP. Thus the MOEA can find multi-solutions of the NES in a single run. Since the MOP is a multi-objective problem in many cases, this paper applies an advanced multi-objective evolutionary algorithm (i.e., Hype algorithm) to solve NES. Our experiment shows better results than or competitive to the four mentioned single-objective optimization in a set of test cases.