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Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. (English) Zbl 0894.90143
Summary: The smoothing Newton method for solving a system of nonsmooth equations $F\left(x\right)=0$, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the $k$th step, the nonsmooth function $F$ is approximated by a smooth function $f\left(·,{\epsilon }_{k}\right)$, and the derivative of $f\left(·,{\epsilon }_{k}\right)$ at ${x}^{k}$ is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if $F$ is semismooth at the solution and $f$ satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is $P$-uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

##### MSC:
 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 65H10 Systems of nonlinear equations (numerical methods) 90C30 Nonlinear programming 49J40 Variational methods including variational inequalities