Construction of proximal distances over symmetric cones. (3rd August 2017)

• Record Type: Journal Article
• Authors: López, Julio; Quiroz, Erik Alex Papa
• Is Part Of: Optimization. Volume 66:Number 8(2017)
• Page Start: 1301
• Page End: 1321
• Publication Date: 2017-08-03
• Subjects: Proximal distance -- spectrally defined function -- symmetric cone programming -- proximal-type algorithms -- primal central paths; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1277998
• Languages: English
• ISSNs: 0233-1934

Abstract

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.