Necessary and sufficient conditions for stable conjugate duality

The conjugate duality, which states that infx∈Xφ(x,0)=maxv∈Y′-φ*(0,v)infx∈Xφ(x,0)=maxv∈Y′-φ*(0,v), whenever a regularity condition on φφ is satisfied, is a key result in convex analysis and optimization, where φ:X×Y→R∪{+∞}φ:X×Y→R∪{+∞} is a convex function, X and Y   are Banach spaces, Y′Y′ is the continuous dual space of Y   and φ*φ* is the Fenchel–Moreau conjugate of φφ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, infx∈X{φ(x,0)+x*(x)}=maxv∈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.