Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new envelope function for nonsmooth DC optimization

Andreas Themelis, Ben Hermans, Panagiotis Patrinos

Abstract

Difference-of-convex (DC) optimization problems are shown to be equivalent to the minimization of a Lipschitz-differentiable "envelope". A gradient method on this surrogate function yields a novel (sub)gradient-free proximal algorithm which is inherently parallelizable and can handle fully nonsmooth formulations. Newton-type methods such as L-BFGS are directly applicable with a classical linesearch. Our analysis reveals a deep kinship between the novel DC envelope and the forward-backward envelope, the former being a smooth and convexity-preserving nonlinear reparametrization of the latter.

A new envelope function for nonsmooth DC optimization

Abstract

Difference-of-convex (DC) optimization problems are shown to be equivalent to the minimization of a Lipschitz-differentiable "envelope". A gradient method on this surrogate function yields a novel (sub)gradient-free proximal algorithm which is inherently parallelizable and can handle fully nonsmooth formulations. Newton-type methods such as L-BFGS are directly applicable with a classical linesearch. Our analysis reveals a deep kinship between the novel DC envelope and the forward-backward envelope, the former being a smooth and convexity-preserving nonlinear reparametrization of the latter.

Paper Structure

This paper contains 15 sections, 35 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Fully parallelizable splitting schemes
  4. Novel smooth DC reformulation
  5. A convexity-preserving nonlinear scaling of the FBE
  6. Paper organization
  7. Notation and known facts
  8. The DC envelope
  9. Connections with the forward-backward envelope
  10. The algorithm
  11. Strongly and hypoconvex functions
  12. Matrix stepsize and relaxation
  13. A parallel three-prox splitting
  14. Simulations
  15. Conclusions

Figures (1)

  • Figure 1: Iteration comparison for random instances of \ref{['eq:PCA']}.

Theorems & Definitions (6)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof