An extension of Chubanov's algorithm to symmetric cones

TL;DR

This paper extends Chubanov's algorithm to symmetric cones, providing a new approach for feasibility problems that improves complexity bounds in cases like semidefinite programming by tracking volume reductions without explicit calculations.

Contribution

It introduces an extension of Chubanov's algorithm to symmetric cones, utilizing volume reduction and spectral norms to improve complexity bounds in feasibility problems.

Findings

Provides concrete upper bounds for minimum eigenvalues.

Achieves better complexity bounds in semidefinite and second order cone programming.

Avoids explicit volume calculations by tracking reductions.

Abstract

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov's method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an earlier work by Kitahara and Tsuchiya on second order cone feasibility problems, progress is measured through the volumes of those intersections: when they become sufficiently small, we know it is time to stop. We never have to explicitly compute the volumes, it is only necessary to keep track of the reductions between iterations. We show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem. Another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that K is decomposed as simple cones. In several key cases, including semidefinite programming and second order cone programming, these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by Pe\~na and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.