“Envelope programming” and conjugate duality

Abstract

In a recent paper D. J. White presented a new approach to the problem of minimizing a differentiable convex function over a convex set. The idea begins with describing the convex function as the envelope of its tangent hyperplanes. With this description the given problem is represented in “min-max” form. An appeal to White's minimax theorem then permits one to interchange the extrema and arrive at a dual problem having “max-min” form. In the present paper White's approach is first generalized and analyzed and then related to well-known results in conjugate duality.