Boundedness and regularity properties of semismooth reformulations of variational inequalities (Q2501118)

The author gives a sufficient condition for boundedness of the level sets of the norm function of a semismooth reformulation of the Karush-Kuhn-Tucker system of a variational inequality problem, when the nonlinear complementarity problem function used in the reformulation is metrically equivalent to the minimum function, and a sufficient and necessary condition when the nonlinear complementarity problem (NCP) function is the minimum function. Main result: The nonsingularity properties [identified by \textit{F. Facchinei, A. Fischer}, and \textit{C. Kanzow}, SIAM J. Optim. 8, No. 3, 850--869 (1998; Zbl 0913.90249)] for a semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, (which is an irrational regular NCP function), hold for the reformulation based on other regular pseudo-smooth NCP functions is precisely showed. Further a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function is proposed. Furthermore, when this function is used to the generalized Newton method (for solving the variational inequality problem), an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only.