%A Chunhui Chen
%A O. L. Mangasarian
%T A Class of Smoothing Functions for Nonlinear and Mixed Complementarity Problems
%D August 1994, Revised October 1994
%R 94-11
%I COMPUTER SCIENCES DEPARTMENT, UNIVERSITY OF WISCONSIN
%C MADISON, WI
%X 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 arbitrarily
accurate solution to
the smooth nonlinear equation as well as the NCP or MCP, is established
for sufficiently large value
of a smoothing parameter $\alpha$. 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 $\alpha$, 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 and Ferris (1993), Harker and Xiao(1990) and Pang and Gabriel (1993).