Partial second-order subdifferentials of -prox-regular functions

Abstract

Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC^{<span style="font-size: small;">2</span>} functions. The class of prox-regular functions covers all convex functions, lower C^{<span style="font-size: small;">2</span>} functions 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.