Zero duality gap conditions via abstract convexity

Abstract

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) $ϵ$-subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems.

Keywords:

• Abstract convexity
• inf-convolution
• Fenchel conjugate
• $ϵ$-subdifferentials sum rule
• zero duality gap

AMS Subject Classifications:

• 52A01
• 47N10
• 49J53
• 49J27
• 49J52
• 90C30

Acknowledgments

The first author is supported by the Australian Research Council through grant IC180100030. The third author is supported by the Australian Research Council, project DP160100854, and benefited from the support of the Conicyt REDES program 180032. We are also grateful to the referees for their constructive comments.

Disclosure statement

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

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by Australian Research Council [DP160100854,IC180100030].