Interior proximal algorithm with variable metric for second-order cone programming: applications to structural optimization and support vector machines

Abstract

In this work, we propose an inexact interior proximal-type algorithm for solving convex second-order cone programs. This kind of problem consists of minimizing a convex function (possibly nonsmooth) over the intersection of an affine linear space with the Cartesian product of second-order cones. The proposed algorithm uses a variable metric, which is induced by a class of positive-definite matrices and an appropriate choice of regularization parameter. This choice ensures the well definedness of the proximal algorithm and forces the iterates to belong to the interior of the feasible set. Also, under suitable assumptions, it is proven that each limit point of the sequence generated by the algorithm solves the problem. Finally, computational results applied to structural optimization and support vector machines are presented.

Keywords:

• proximal method
• second-order cone programming
• variable metric
• structural optimization
• multiload model
• support vector machines
• robust classifier

Acknowledgements

The authors are grateful to the anonymous referees for their comments on the original manuscript. Their suggestions were very useful to improve this work. The authors also wish to thank Professor Paulo J.S. Silva for stimulating conversations and helpful comments. This work was partially supported by FONDECYT under grants 1050706 and 1070297, FONDAP in Applied Mathematics, BASAL project, and the Millennium Scientific Institute on Complex Engineering Systems funded by MIDEPLAN-Chile. The second author was also supported by CONICYT.