A Cubic Regularization Method For Solving Nonsmooth Equations

Nonsmooth equations are widely used in the field of optimization which provides a uni-fied framework for studying many problems.The system of nonsmooth equations arises from many applications,including nonlinear complementarity problems,variational inequality prob-lems,KKT systems arising from optimization problems,etc.In recent years,a class of cubic regularization methods for solving smooth equations have attracted great attention.In each iteration,the iterative step is obtained by solving the minimum point of a cubic regularization function.It is proved that the method can converge to the second-order stationary point.More importantly,the cubic regularization method has better worst-case complexity than the second order method,and satisfactory numerical performance.Based on the above discussion,the aim of this paper is to study a cubic regularization method for solving nonsmooth equations.A cubic regularization method is proposed for solving a nonlinear system of semismooth equations.By adapting the cubic regularization algorithm for solving the smooth equations,we give the cubic regularization algorithm for nonsmooth equations.By combing the classical trust region technique,the proposed method is ensured to be globally convergent.When the algorithm satisfies the Cauchy con diton,the iterates converge to first-order critical point.In order to prove the local convergence rate of the algorithm,we need to solve the subproblem inexactly and some stopping criterion is required.Under the BD-regular condition,we prove that the cubic regularizatin method is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations.Finally,we use the cubic regularization algorithm to solve nonlinear complementarity problems,and the efficiency of our method is verified by numerical results.