Strong Convergence of a Two-Step Modified Newton Method for Weighted Complementarity Problems

Abstract

This paper focuses on the weighted complementarity problem (WCP), which is widely used in the fields of economics, sciences and engineering. Not least because of its local superlinear convergence rate, smoothing Newton methods have widespread application in solving various optimization problems. A two-step smoothing Newton method with strong convergence is proposed. With a smoothing complementary function, the WCP is reformulated as a smoothing set of equations and solved by the proposed two-step smoothing Newton method. In each iteration, the new method computes the Newton equation twice, but using the same Jacobian, which can avoid consuming a lot of time in the calculation. To ensure the global convergence, a derivative-free line search rule is inserted. At the same time, we develop a different term in the solution of the smoothing Newton equation, which guarantees the local strong convergence. Under appropriate conditions, the algorithm has at least quadratic or even cubic local convergence. Numerical experiments indicate the stability and effectiveness of the new method. Moreover, compared to the general smoothing Newton method, the two-step smoothing Newton method can significantly improve the computational efficiency without increasing the computational cost.