Generalized Moreau–Rockafellar results for composed convex functions

Abstract

We give two generalized Moreau–Rockafellar-type results for the sum of a convex function with a composition of convex functions in separated locally convex spaces. Then we equivalently characterize the stable strong duality for composed convex optimization problems through two new regularity conditions, which also guarantee two formulae of the subdifferential of the mentioned sum of functions. We also treat some special cases, rediscovering older results in the literature. A discussion on the topological assumptions for the vector function used in the composition closes this article.

Keywords:

• conjugate functions
• Moreau–Rockafellar results
• regularity conditions
• stable strong duality
• composed convex functions

AMS Subject Classifications::

• 90C25
• 49N15
• 65K10

Acknowledgements

This research was partially supported by DFG (German Research Foundation), project WA 922/1–2. The authors are grateful to an anonymous reviewer for valuable comments and remarks.