A Note on Nondifferentiable Symmetric Duality

Under suitable hypotheses on the function / , the two constrained minimization problems: MIN/ fyy subject to x > 0, -fy > 0; MAX/ fxx subject to y > 0, fx > 0; are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to: MINf(x,y) py subject to x e T,-p e S* n dy(-f)(x,y), in which T and S are convex cones, S* is the dual cone of S, and dy denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.