Construction of proximal distances over symmetric cones

Julio López, Erik Alex Papa Quiroz

Research output: Contribution to journalArticlepeer-review

Abstract

This paper is devoted to the study of proximal distances defined over symmetric cones, which include the non-negative orthant, the second-order cone and the cone of positive semi-definite symmetric matrices. Specifically, our first aim is to provide two ways to build them. For this, we consider two classes of real-valued functions satisfying some assumptions. Then, we show that its corresponding spectrally defined function defines a proximal distance. In addition, we present several examples and some properties of this distance. Taking into account these properties, we analyse the convergence of proximal-type algorithms for solving convex symmetric cone programming (SCP) problems, and we study the asymptotic behaviour of primal central paths associated with a proximal distance. Finally, for linear SCP problems, we provide a relationship between the proximal sequence and the primal central path.

Original languageEnglish
Pages (from-to)1301-1321
Number of pages21
JournalOptimization
Volume66
Issue number8
DOIs
StatePublished - 3 Aug 2017

Bibliographical note

Funding Information:
This research was supported by CONICYT-Chile, via FONDECYT [project 1160894] (Julio L?pez); Postdoctoral Scholarship CAPES-FAPERJ Edital [PAPD-2011] (Erik Alex Papa).

Publisher Copyright:
© 2017 Informa UK Limited, trading as Taylor & Francis Group.

Keywords

  • Proximal distance
  • primal central paths
  • proximal-type algorithms
  • spectrally defined function
  • symmetric cone programming

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