The locally Chen-Harker-Kanzow-Smale smoothing functions for mixed complementarity problems.

*(English)*Zbl 1423.90260Summary: 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.

##### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C30 | Nonlinear programming |

##### Keywords:

mixed complementarity problems; smoothing projection functions; semismooth; smoothing Newton method
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\textit{Z. Zhou} and \textit{Y. Peng}, J. Glob. Optim. 74, No. 1, 169--193 (2019; Zbl 1423.90260)

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