On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions
M. D. Fajardo, J. Vidal-Nunez
TL;DR
The article develops a subdifferential defined through a generalized $c$-conjugation for evenly convex functions and studies its relation to the $c$-conjugate domain and the $ε$-directional derivative. It extends classical DC optimization results by formulating optimality conditions for problems of the form $f-g$ where $f$ and $g$ are (evenly) convex, using the $c$-subdifferential and its $ε$-variants. Key contributions include characterizations of the $c$-subdifferential, links to e-convex hulls, and HU-type necessary optimality conditions expressed via $c$-subdifferentials, along with a discussion of their limitations and potential applications to Toland–Singer duality. The framework generalizes Fenchel-based tools to the broader setting of evenly convex functions, offering a path toward non-smooth DC analysis under generalized convexity. Future work includes deeper duality results and separation-type results for the $c$-conjugation scheme.
Abstract
This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the ε-directional derivative. In addition, we also present necessary conditions for ε-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions will be written via this generalized notion of subdifferential studied in the first sections of the paper.