New regularity conditions and Fenchel dualities for DC optimization problems involving composite functions

Abstract

We consider the following DC composite optimization problem (P): $\underset{x\in X}{inf}\left\{f_1\left(x\right)-f_2\left(x\right)+g_1\left(h\right(x\left)\right)-g_2\left(h\right(x\left)\right)\right\},$ where $f_1,f_2$ and $g_1,g_2$ are proper convex functionals defined on locally convex Hausdorff topological vector spaces X and Y respectively, and h is a proper K-convex mapping from X to Y. By using the properties of the epigraph of the conjugate functions, we introduce some new regularity conditions and obtain complete characterizations for weak / strong / stable Fenchel dualities and for zero / stable zero duality gap properties of problem (P).

Keywords:

• Regularity condition
• strong duality
• zero duality gap property
• composite optimization problem
• DC programming

2010 Mathematics Subject Classifications:

• 90C26
• 90C46

Acknowledgements

Authors are grateful to the referees and handling editor for their valuable suggestions and comments which improved the previous draft of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

In this research, first author was supported in part by the National Natural Science Foundation of China [grant number 11861033] and the Scientific Research Fund of Hunan Provincial Education Department [grant number 17A172].