A Weighted Biobjective Transformation Technique for Locating Multiple Optimal Solutions of Nonlinear Equation Systems

Due to the fact that a nonlinear equation system (NES) may contain multiple optimal solutions, solving NESs is one of the most important challenges in numerical computation. When applying evolutionary algorithms to solve NESs, two issues should be considered: 1) how to transform an NES into a kind of optimization problem and 2) how to develop an optimization algorithm to solve the transformed optimization problem. In this paper, we tackle the first issue by transforming an NES into a weighted biobjective optimization problem. By the above transformation, not only do all the optimal solutions of an original NES become the Pareto optimal solutions of the transformed biobjective optimization problem, but also their images are different points on a linear Pareto front in the objective space. In addition, we suggest an adaptive multiobjective differential evolution, the goal of which is to effectively locate the Pareto optimal solutions of the transformed biobjective optimization problem. Once these solutions are found, the optimal solutions of the original NES can also be obtained correspondingly. By combining the weighted biobjective transformation technique with the adaptive multiobjective differential evolution, we propose a generic framework for the simultaneous locating of multiple optimal solutions of NESs. Comprehensive experiments on 38 NESs with various features have demonstrated that our framework provides very competitive overall performance compared with several state-of-the-art methods.