A Class of Smoothing Functions for Nonlinear and Mixed Complementarity Problems

Abstract

We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)+ =max {0,x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrary accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for sufficiently large value of smoothing parameter ?. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter ?, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with those for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse & Ferris [8], Harker & Xiao [13] and Pang & Gabriel [30].