Partial second-order subdifferentials of -prox-regular functions

Subject Areas : Statistics

S. Nadi ¹ (Ph.D. student, Department of Mathematics, Sahand University of Technology, Tabriz, Iran)
J. Vakili ² (Department of Applied Mathematics, School of Mathematics Science, University of Tabriz, Tabriz, Iran)

Received: 2017-04-04 Accepted : 2017-07-11 Published : 2017-05-01

Keywords: توابع تقریباً-منظم, نگاشت یکنوای ماکسیمال, زیرمشتقات جزئی مرتبه دوم, هم‌مشتق, آنالیز تغییرات,

Abstract :

Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2functions. The class of prox-regular functions covers all convex functions, lower C2functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regular functions have been developed in Banach and Hilbert spaces. In this paper, the parametric prox-regular functions are defined using limiting subdifferentials. Also, a partial second-order subdifferential is defined here for extended real valued functions of two variables corresponding to its variables through coderivatives of first-order partial subdifferential mappings. Then, relations between maximal monotonicity of the partial first-order subdifferentials of these functions and the positive-semidefiniteness of the coderivatives of partial first order subdifferential mapping are investigated. Finally, we present necessary and sufficient conditions for ∂ -prox-regular functions to be convex based on positive-semidefiniteness of the partial second-order subdifferentials mappings.