A closedness condition and its applications to DC programs with convex constraints

Abstract

This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called ‘Toland–Fenchel–Lagrange’ duality theorems. As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived. Then, we establish optimality conditions for DC problem under convex constraints. Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well. Most of the results are established under the (CC) condition. This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs.

Keywords::

• DC programs
• closedness conditions
• closed-cone constraint qualification
• Farkas lemmas
• Fenchel–Lagrange duality
• Toland–Fenchel–Lagrange duality

AMS Classifications::

• 90C25
• 90C26
• 90C46
• 49K30

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper. Parts of the work of N. Dinh were realized during his visit at the Laboratory of Applied Mathematics, University of PAU to which he would like to express his sincere thanks for the hospitality he received.