Department of Applied Mathematics & Physics, Kyoto Univiversity

Technical Report #99001 (February, 1999)

Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems
by Donghui Li and Masao Fukushima

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. The smoothing Newton method converges globally if the problem involves a differentiable $P_0$ function. Under suitable conditions, the method exhibits a quadratic convergence property. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.