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.
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading and helpful suggestions that improved the paper greatly.
Notes
No potential conflict of interest was reported by the authors.
1 Recall that a second-order cone [Citation37] is defined by the set .
2 Recall that the -subdifferential of a
at x is defined by
, for some
, and the subdifferential by
.