Necessary and sufficient conditions for stable conjugate duality

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.

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