Interior proximal bundle algorithm with variable metric for nonsmooth convex symmetric cone programming

Abstract

This paper is devoted to the study of a bundle proximal-type algorithm for solving the problem of minimizing a nonsmooth closed proper convex function subject to symmetric cone constraints, which include the positive orthant in , the second-order cone, and the cone of positive semidefinite symmetric matrices. On the one hand, the algorithm extends the proximal algorithm with variable metric described by Alvarez et al. to our setting. We show that the sequence generated by the proposed algorithm belongs to the interior of the feasible set by an appropriate choice of a regularization parameter. Also, it is proven that each limit point of the sequence generated by the algorithm solves the problem. On the other hand, we provide a natural extension of bundle methods for nonsmooth symmetric cone programs. We implement and test numerically our bundle algorithm with some instances of Euclidean Jordan algebras.

Keywords:

• Euclidean Jordan algebra
• symmetric cone programming
• variable metric
• proximal method
• bundle methods

Acknowledgements

The authors wish to thank Professor Claudia Sagastizábal who provided us the codes of the bundle algorithm described in [45].

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by FONDECYT under [grant number 1160894], [grant number 1130176]; Millennium Scientific Institute on Complex Engineering Systems [grant number ICM: P-05-004-F], (CONICYT: FBO16). Finally, we thank the anonymous referees for their careful reading and helpful suggestions that improved the paper greatly.