Envelope Functions: Unifications and Further Properties

Recently, the forward-backward and Douglas-Rachford envelope functions were proposed in the literature. The stationary points of these envelope functions have a close relationship with the solutions of the possibly nonsmooth optimization problem to be solved. The envelopes were shown to be smooth and convex under some additional assumptions. Therefore, these envelope functions create powerful bridges between nonsmooth and smooth optimization. In this paper, we present a general envelope function that unifies and generalizes these envelope functions. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present an envelope function for the generalized alternating projections method (GAP), named the GAP envelope. It enables for convex feasibility problems with two sets, of which one is affine, to be solved by finding any stationary point of the smooth and under some assumptions convex GAP envelope.