Zero duality gap conditions via abstract convexity
Files
(Published version)
Date
2022
Authors
Bui, H.T.
Burachik, R.S.
Kruger, A.Y.
Yost, D.T.
Editors
Advisors
Journal Title
Journal ISSN
Volume Title
Type:
Journal article
Citation
Optimization, 2022; 71(4):811-847
Statement of Responsibility
Conference Name
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) (Formula presented.) -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
School/Discipline
Dissertation Note
Provenance
Description
Access Status
Rights
Copyright 2021 Informa UK Limited, trading as Taylor & Francis Group
Access Condition Notes: Accepted manuscript available after 1 July 2022