Smoothing Entries Updating Newton-like Methods For Nonsmooth Equations

The numerical solution of nonsmooth equation is an important research topic in computational mathematics and mathematical programming, which provides a unified framework for studying many problems in mathematical programming. The system of nonsmooth equations arises from many applications. Pang Jong-shi and Qi Liqun reviewed eight problems in the study of complementarity problems, variational inequality problems and optimization problems, which can be reformulated as systems of nonsmooth equations. In recent years, the international and domestic scholars pay great attention to numerical solution of nonsmooth equation, especially on complementarity problems and variational inequality problems. The plentiful research results about projection, generalized Newton method and smoothing Newton method laid a theoretical foundation and provide a lot of effective and practical methods for numerical solution of nonsmooth equation.In fact, we often encounter sparse nonsmooth equations, that is, the Jacobi matrix or generalized Jocobi matrix of nonsmooth mapping is sparse. How to solve such problems effectively use of sparity is a worthy of further consideration issue.The entries update Newton-like methods are effective iterative methods for smooth nonlinear equations and smooth unconstrained optimization problems. They can calculate approximate Jacobi matrixes or Hesse matrixes as few as possible and have convergence rate between one and two. Therefore they are applicable for solving sparse large-scale problems.We consider these methods to solve sparse nonsmooth equation here. First, we give the smoothing entries update Newton-like methods algorithm and theoretical analysis, and then we weaken the theorem conditions, give the mixed Newton and entries update Newton-like methods. Locally superlinear result is presented under the assumption of weaker condition. We illustrate the smoothing method on box constrained variational inequalities problem. Preliminary numerical tests show these methods are efficient.