On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima

Research output: Contribution to journalArticlepeer-review

16 Citations (SciVal)

Abstract

We provide theory for the computation of convex envelopes of non-convex functionals including an ℓ2-term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the ℓ2-term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional, we necessarily have a direction in which the convex envelope is affine.

Original languageEnglish
Pages (from-to)66-84
JournalJournal of Optimization Theory and Applications
Volume183
Issue number1
Early online date2019 May 29
DOIs
Publication statusPublished - 2019

Subject classification (UKÄ)

  • Mathematics

Keywords

  • Convex envelope
  • Fenchel conjugate
  • Non-convex/non-smooth optimization
  • Proximal hull
  • Regularization

Fingerprint

Dive into the research topics of 'On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima'. Together they form a unique fingerprint.

Cite this