Duality in nonconvex optimization

We examine a notion of duality which appears to be useful in situations where the usual convex duality theory is not appropriate because the functional to be minimized is not convex. The principle is a generalization of a duality theorem derived previously (J. F. Toland, University of Essex Fluid Mechanics Research Institute, Report No. 77, November 1976, Arch. Rational Mech. Analysis (in press)), for nonconvex problems. The generalization is considerable, since no assumptions are made on the functional to be minimized, other than that it can be embedded in a family of perturbed problems. If such an embedding is possible, then the main theorem depends only on some rather well-known results in the theory of conjugate convex functions. We develop all the previously derived abstract results in this more general framework. The earlier work is seen to be a special case of this generalized duality theory. We treat abstract problems which are typical of those arising in the calculus of variations, and some applications are considered.