On the Convergence of Gauss-type Proximal Point Method for Smooth Generalized Equations

Abstract

Let X and Y be Banach spaces and Ω be an open subset of X. Let f : X → Y be a Frechet differentiable function on Ω and F : X 2Y be a set valued mapping with closed graph. We deal with smooth generalized equations which is de ned by the sum of Frechet di erentiable function and a set valued mapping. Under some sufficient conditions, a Gauss-type proximal point algorithm (G-PPA) is introduced and studied for solving generalized equations of the form 0 ∈ 2 f(x) + F(x). Indeed, when F is metrically regular we analyze semi-local and local convergence of the G-PPA. Furthermore, we give a numerical example to justify the convergence results of the G-PPA.