Dualities for non-Euclidean smoothness and strong convexity under the light of generalized conjugacy

Relative smoothness and strong convexity have recently gained considerable attention in optimization. These notions are generalizations of the classical Euclidean notions of smoothness and strong convexity that are known to be dual to each other. However, conjugate dualities for non-Euclidean relative smoothness and strong convexity remain an open problem as noted earlier by Lu, Freund and Nesterov [SIAM J. Optim., 28 (2018), pp. 333-354]. In this paper we address this question by introducing the notions of anisotropic strong convexity and smoothness as the respective dual counterparts. The dualities are developed under the light of generalized conjugacy which leads us embed the anticipated dual notions within the superclasses of certain upper and lower envelopes. In contrast to the Euclidean case these inclusions are proper in general as showcased by means of counterexamples.