caa a22 4500 47576921X CHVBK 20180406123615.0 cr unu---uuuuu 170329e20000101xx s 000 0 eng 10.1007/s101079900127 doi (NATIONALLICENCE)springer-10.1007/s101079900127 A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities [Elektronische Daten] [Liqun Qi, Defeng Sun, Guanglu Zhou] 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. Springer-Verlag Berlin Heidelberg, 2000 Key words: variational inequalities - nonsmooth equations - smoothing approximation - smoothing Newton method - convergence Mathematics Subject Classification (1991): 90C33, 90C30, 65H10 nationallicence Qi Liqun 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 aut Sun Defeng 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 aut Zhou Guanglu 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 aut https://doi.org/10.1007/s101079900127 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s101079900127 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Qi Liqun 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 aut NATIONALLICENCE 700 1- Sun Defeng 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 aut NATIONALLICENCE 700 1- Zhou Guanglu 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 aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer