Necessary and sufficient conditions for stable conjugate duality

By Regina Sandra Burachik, Vaithilingam Jeyakumar, Zhiyou Wu. 2006

65 citations recorded by the source corpus.

Abstract The conjugate duality, which states that inf x ∈ X φ ( x , 0 ) = max v ∈ Y ′ - φ * ( 0 , v ) , whenever a regularity condition on φ is satisfied, is a key result in convex analysis and optimization, where φ : X × Y → R ∪ { + ∞ } is a convex function, X and Y are Banach spaces, Y ′ is the continuous dual space of Y and φ * is the Fenchel–Moreau conjugate of φ . In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf x ∈ X { φ ( x , 0 ) + x * ( x ) } = max v ∈ Y ′ { - φ * ( - x * , v ) } , ∀ x * ∈ X ′ and then obtain a new epigraph regularity condition for the conjugate duality. The regularity condition is shown to be much more general than the popularly known interior-point type conditions. As an easy consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel–Rockafellar duality theorem. In the case where one of the functions involved is a polyhedral convex function, we provide generalized interior-point conditions for the epigraph regularity condition. Moreover, we show that a stable Fenchel's duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/451639510d316e72a5e745205b992ed9a3de2cf4

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Citations in the corpus

Regularity conditions via generalized interiority notions in convex optimization: New achievements and their relation to some classical statements (2009, background): “Remark 12 (a) Let us notice that condition (7) is referred in the literature as stable strong duality (see [8,15,42] for more details) and obviously guarantees strong duality for (PF) (DF).”
A closedness condition and its applications to DC programs with convex constraints (2010, methodology): “Moreover, in the cases where the cost functional is not continuous at any point in the feasible set, another condition, often called the closedness condition [10,11,13,14], should be imposed.”
Generalized Moreau–Rockafellar results for composed convex functions (2009, background): “g From these two theorems we can get the following statement, given also in [5].”