Prox-regular functions are variationally s-convex or have tilt-stable minimizers exactly when their quadratic bundles satisfy a uniform quadratic lower bound, without requiring subdifferential continuity.
Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis
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In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.
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Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles
Prox-regular functions are variationally s-convex or have tilt-stable minimizers exactly when their quadratic bundles satisfy a uniform quadratic lower bound, without requiring subdifferential continuity.