Structured Sparsity Promoting Functions

Lixin Shen, Bruce W. Suter, Erin E. Tripp

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Motivated by the minimax concave penalty-based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we provide sparsity guarantees for the general family of functions. We further study the behavior of the proximity operators of several special functions, including indicator functions of closed and convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases.

Original languageEnglish (US)
Pages (from-to)386-421
Number of pages36
JournalJournal of Optimization Theory and Applications
Issue number2
StatePublished - Nov 1 2019


  • Moreau envelope
  • Proximity operator
  • Sparsity
  • Thresholding operator
  • Variable selection

ASJC Scopus subject areas

  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics


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