Envelope Functions: Unifications and Further Properties

TL;DR

This paper introduces a unified envelope function framework that generalizes existing envelopes like forward-backward and Douglas-Rachford, providing new properties and a novel envelope for the generalized alternating projections method.

Contribution

It presents a general envelope function unifying previous envelopes and introduces the GAP envelope for convex feasibility problems, enhancing solution methods.

Findings

Unified envelope function generalizes existing envelopes.

Properties of the general envelope sharpen known results.

GAP envelope enables solving convex feasibility problems via stationary points.

Abstract

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.