Newton–Steffensen–Type Method for Perturbed Nonsmooth Subanalytic Variational Inequalities
Keywords:
Aubin–like property, variational inclusions, subanalytic function, Newton’s method, Steffensen’s method, local convergence, Clarke’s subdifferential, Holder/center– ¨ Holder condition, divided difference, set–valued mapAbstract
This paper is devoted to Newton–Steffensen–type method for approximating the unique solution of perturbed nonsmooth subanalytic variational inclusion in finite–dimensional spaces. We use a combination of Newton’s method studied by Bolte et al. [14] for locally Lipschitz subanalytic function in order to solve nonlinear equations, with Steffensen’s method [1, 2, 3, 9, 19]. Using the Lipschitz–like concept of set–valued mappings, the subanalyticity hypothesis on the involved function and some condition on divided difference operator, the superlinear convergence is established. We also present a finer convergence analysis using some ideas introduced by us in [4, 7, 8] for nonlinear equations. Finally, we present some new regula–falsi–type method for set–valued map.
