A Weighted Biobjective Transformation Technique for Locating Multiple Optimal Solutions of Nonlinear Equation Systems
IEEE Transactions on Evolutionary Computation
Due to the fact that a nonlinear equation system may contain multiple optimal solutions, solving nonlinear equation systems is one of the most important challenges in numerical computation. When applying evolutionary algorithms to solve nonlinear equation systems, two issues should be considered: i) how to transform a nonlinear equation system into a kind of optimization problem, and ii) how to develop an optimization algorithm to solve the transformed optimization problem. In this paper, we
... kle the first issue by transforming a nonlinear equation system into a weighted biobjective optimization problem. By the above transformation, not only do all the optimal solutions of an original nonlinear equation system 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 nonlinear equation system 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 nonlinear equation systems. Comprehensive experiments on 38 nonlinear equation systems with various features have demonstrated that our framework provides very competitive overall performance compared with several state-ofthe-art methods.