Abstract
This article uses classical notions of convex analysis over Euclidean spaces, like Gale & Klee’s boundary rays and asymptotes of a convex set, or the inner aperture directions defined by Larman and Brøndsted for the same class of sets, to provide a generalization of the Clark–Duffin Theorem.
On this ground, we are able to characterize objective functions and, respectively, feasible sets for which the duality gap is always zero, regardless of the value of the constraints and, respectively, of the objective function.
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Notes
The naive proof of the Clark–Duffin Theorem comes from a reviewer report to a previous version of this paper.
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Acknowledgements
The authors are grateful to the two anonymous referees for their helpful comments and suggestions which have been included in the final version of the paper.
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Communicated by Jean-Pierre Crouzeix.
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Ernst, E., Volle, M. Zero Duality Gap for Convex Programs: A Generalization of the Clark–Duffin Theorem. J Optim Theory Appl 158, 668–686 (2013). https://doi.org/10.1007/s10957-013-0287-7
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DOI: https://doi.org/10.1007/s10957-013-0287-7