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The paper is devoted to the study, characterizations, and applications ofvariational convexity of functions, the property that has been recentlyintroduced by Rockafellar together with its strong counterpart. First we showthat these variational properties of an extended-real-valued function areequivalent to, respectively, the conventional (local) convexity and strongconvexity of its Moreau envelope. Then we derive new characterizations of bothvariational convexity and variational strong convexity of general functions viatheir second-order subdifferentials (generalized Hessians), which arecoderivatives of subgradient mappings. We also study relationships of thesenotions with local minimizers and tilt-stable local minimizers. The obtainedresults are used for characterizing related notions of variational and strongvariational sufficiency in composite optimization with applications tononlinear programming.

Variational Convexity of Functions and Variational Sufficiency in Optimization