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.
Bibliographical noteFunding Information:
This research was supported by CONICYT-Chile, via FONDECYT [project 1160894] (Julio López); Postdoctoral Scholarship CAPES-FAPERJ Edital [PAPD-2011] (Erik Alex Papa).
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- Proximal distance
- primal central paths
- proximal-type algorithms
- spectrally defined function
- symmetric cone programming