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A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $$P_0$$ function. (English) Zbl 1225.90140
Summary: We propose a new class of smoothing functions which uniformly approximates the median function of three scalars. The proposed functions are the generalization of the smoothing function proposed by Li and Fukushima. Some favorable properties of the functions are investigated. By using the proposed functions, we reformulate the box constrained variational inequality problem (VIP) as a system of parameterized smooth equations, and then propose a smoothing Newton algorithm with a nonmonotone line search to solve the VIP. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results for test problems from MCPLIB are also reported, which demonstrate that the proposed smoothing functions are valuable and the proposed algorithm is effective.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 65K10 Numerical optimization and variational techniques
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