A comparison of some recent regularity conditions for Fenchel duality

This article provides an overview on regularity conditions for Fenchel duality in convex optimization. Our attention is focused, on the one hand, on three generalized interior-point regularity conditions expressed by means of the quasi interior and of the quasi-relative interior and, on the other hand, on two closednesstype conditions that have been recently introduced in the literature. We discuss how they do relate to each other, but also to several other classical ones and illustrate these investigations by numerous examples.