NDLI: Strong Duality in Robust Convex Programming: Complete Characterizations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Jeyakumar, V. Li, G. Y.
Copyright Year	2010
Abstract	Duality theory has played a key role in convex programming in the absence of data uncertainty. In this paper, we present a duality theory for convex programming problems in the face of data uncertainty via robust optimization. We characterize strong duality between the robust counterpart of an uncertain convex program and the optimistic counterpart of its uncertain Lagrangian dual. We provide a new robust characteristic cone constraint qualification which is necessary and sufficient for strong duality in the sense that the constraint qualification holds if and only if strong duality holds for every convex objective function of the program. We further show that this strong duality always holds for uncertain polyhedral convex programming problems by verifying our constraint qualification, where the uncertainty set is a polytope. We derive these results by way of first establishing a robust theorem of the alternative for parameterized convex inequality systems using conjugate analysis. We also give a convex characteristic cone constraint qualification that is necessary and sufficient for strong duality between the deterministic dual pair: the robust counterpart and its Lagrangian dual. Through simple numerical examples we also provide an insightful account of the development of our duality theory.
Starting Page	3384
Ending Page	3407
Page Count	24
File Format	PDF
ISSN	10526234
DOI	10.1137/100791841
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	6
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-12-02
Access Restriction	Subscribed
Subject Keyword	Lagrangian duality Optimality conditions, duality characteristic cones convex programming under uncertainty robust optimization Nonconvex programming, global optimization strong duality Nonlinear programming Quadratic programming
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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