Envelope Functions: Unifications and Further Properties

Research output: Contribution to journalArticle


Forward–backward and Douglas–Rachford splitting are methods for structured nonsmooth optimization. With the aim to use smooth optimization techniques for nonsmooth problems, the forward–backward and Douglas–Rachford envelopes where recently proposed. Under specific problem assumptions, these envelope functions have favorable smoothness and convexity properties and their stationary points coincide with the fixed-points of the underlying algorithm operators. This allows for solving such nonsmooth optimization problems by minimizing the corresponding smooth convex envelope function. In this paper, we present a general envelope function that unifies and generalizes existing ones. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present a new interpretation of the underlying methods as being majorization–minimization algorithms applied to their respective envelope functions.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematical Analysis


  • Envelope functions, First-order methods, Large-scale optimization, Nonsmooth optimization, Smooth reformulations
Original languageEnglish
Pages (from-to)673-698
JournalJournal of Optimization Theory and Applications
Issue number3
Early online date2018 Jun 12
Publication statusPublished - 2018
Publication categoryResearch