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In this paper, based on the 3-1 piecewise nonlinear complementarity problem (NCP) function, we proposed a smoothing Newton-type method for the generalized nonlinear complementarity problem (GNCP) w...

In this paper, based on the Kanzow-Kleinmichel (KK) function, we introduce a new nonlinear complementarity problem (NCP) function: penalized KK function. We show that the function possesses desirable properties similar to those of the KK function. Based on this new NCP function, we reformulate the NCP to a system of semismooth equations. We also propose a new semismooth Levenberg–Marquardt method to solve the system of semismooth equations that employs both trust region techniques and line searches. The global and quadratic convergence properties can be established under very mild assumptions. Numerical results show the effectiveness of the proposed algorithm and also indicate that superior behaviour of the proposed new NCP function.

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P 0 function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.

Based on the Logarithmic-Quadratic Proximal (LQP) method 2, we propose a new prediction-correction method for nonlinear complementarity problem (NCP). We obtain the predictor through a simplified inexact logarithmic-quadratic proximal method under a relaxed inexact criterion. The corrector is obtained by the improved extragradient method. Under certain conditions, the global convergence of the proposed method is proved. Preliminary numerical results indicate the efficiency of the proposed method.

This paper provides an analysis of the iterative complexity of a predictor-corrector type interior-point algorithm for a class of non-monotone nonlinear complementarity problems, i.e., the nonlinear P∗-complementarity problems, which is quite general because it includes as a special case the monotone complementarity problem. At each corrector step, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. The proof of the iterative complexity of the proposed algorithm requires that the mapping associated the problem satisfies a scaled Lipschitz condition.