A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities |
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Authors: | Liqun Qi Defeng Sun Guanglu Zhou |
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Institution: | School of Mathematics, The University of New South Wales, Sydney 2052, Australia?e-mail: L.Qi@unsw.edu.au,?sun@maths.unsw.edu.au,?zhou@maths.unsw.edu.au, AU
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Abstract: | In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the
box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions,
we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent
nonsmooth equation H(u,x)=0, where H:ℜ
2n
→ℜ
2n
, u∈ℜ
n
is a parameter variable and x∈ℜ
n
is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z
k
=(u
k
,x
k
)} such that the mapping H(·) is continuously differentiable at each z
k
and may be non-differentiable at the limiting point of {z
k
}. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which
play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not
require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if
we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the
proposed methods for particularly chosen smoothing functions are very promising.
Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999 |
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Keywords: | : variational inequalities – nonsmooth equations – smoothing approximation – smoothing Newton method – convergence Mathematics Subject Classification (1991): 90C33 90C30 65H10 |
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