Polar convolution

Michael P. Friedlander, Ives MacEdo, Ting Kei Pong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)


The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness, and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.

Original languageEnglish
Pages (from-to)1366-1391
Number of pages26
JournalSIAM Journal on Optimization
Issue number2
Publication statusPublished - 16 May 2019


  • Gauge optimization
  • Max convolution
  • Proximal algorithms

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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