Introduced by Rockafellar in 2019, variational convexity is a generalized convexity with emerging interest, owing to its power to reduce nonconvex problems to convex ones locally.

In this talk, I will showcase recent results on variational convexity, which include, among other things, a complete characterization of local nonexpansive-type properties of proximal operators. The level proximal subdifferentials of prox-bounded functions play a central role in this process. Aided by these results, several celebrated first-order algorithms are shown to exhibit convex-like behavior when the iterates are "close enough" in a primal-dual sense. Practical strategies for getting "close enough" will be provided. To explore the richness of the class of variationally convex functions, I will present several operations that preserve variational convexity. Open problems regarding variational convexity will also be discussed.