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For the existence of strong duality in convex optimization regularityconditions play an indisputable role. We mainly deal in this paper withregularity conditions formulated by means of different generalizations of thenotion of interior of a set. The primal-dual pair we investigate is a generalone expressed in the language of a perturbation function and by employing itsFenchel-Moreau conjugate. After providing an overview on the generalizedinterior-point conditions that exist in the literature we introduce several newones formulated by means of the quasi interior and quasi-relative interior. Weunderline the advantages of the new conditions vis-\'a-vis the classical onesand illustrate our investigations by numerous examples. We close the paper byparticularizing the general approach to the classical Fenchel and Lagrangeduality concepts.

Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements