# The locally Chen–Harker–Kanzow–Smale smoothing functions for mixed complementarity problems

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## Abstract

According to the structure of the projection function onto the box set \(\varPi _X\) and the Chen–Harker–Kanzow–Smale (CHKS) smoothing function, a new class of smoothing projection functions onto the box set are proposed in this paper. The new smoothing projection functions only smooth \(\varPi _X\) in neighborhoods of nonsmooth points of \(\varPi _X\), and keep unchanged with \(\varPi _X\) at other points, hence they are referred as the locally Chen–Harker–Kanzow–Smale (LCHKS) smoothing functions. Based on the Robinson’s normal equation and the LCHKS smoothing functions, a smoothing Newton method with its convergence results is proposed for solving mixed complementarity problems. Compared with smoothing Newton methods based on various smoothing projection functions, the computations of the LCHKS smoothing function, the function value and its Jacobian matrix of the Newton equation become cheaper, and the Newton direction can be found by solving a low dimensional linear equation, hence the smoothing Newton method based on the LCHKS smoothing functions shows more efficient for large-scale mixed complementarity problems. The LCHKS smoothing functions are proved to be feasible, continuously differentiable, uniform approximations of \(\varPi _X\), globally Lipschitz continuous and strongly semismooth, which are important to establish the superlinear and quadratic convergence of the smoothing Newton method. The proposed smoothing Newton method is implemented in MATLAB and numerical tests are done on the MCPLIB test collection. Numerical results show that the smoothing Newton method based on the LCHKS smoothing functions is promising for mixed complementarity problems.

## Keywords

Mixed complementarity problems Smoothing projection functions Semismooth Smoothing Newton method## Notes

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