Necessary and sufficient conditions for stable conjugate duality

The conjugate duality, which states that inf(x is an element of X) phi(x, 0)=max(upsilon is an element of Y') - phi*(0, upsilon), whenever a regularity condition on phi is satisfied, is a key result in convex analysis and optimization, where phi: X x Y -> R boolean OR {+infinity} is a convex function, X and Y are Banach spaces, Y' is the continuous dual space of Y and phi* is the Fenchel-Moreau conjugate of phi. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf(x is an element of X) {phi(x, 0) + x*(x)} = max(upsilon is an element of Y') {-phi*(-x*, upsilon)}, for all x* is an element of 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. (c) 2005 Elsevier Ltd. All rights reserved.