# An extension of Chubanov’s algorithm to symmetric cones

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## Abstract

In this work we present an extension of Chubanov’s algorithm to the case of homogeneous feasibility problems over a symmetric cone \( {\mathcal {K}}\). 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 \( {\mathcal {K}}\). 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 \( {\mathcal {K}}\) 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 Peña and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.

## Keywords

Symmetric cone Feasibility problem Chubanov’s method## Mathematics Subject Classification

90C25 90C22 65K05## Notes

### Acknowledgements

We thank the referees for their helpful and insightful comments, which helped to improve the paper. This article benefited from an e-mail discussion with Prof. Javier Peña, which helped clarify some points regarding [23]. T. Kitahara is supported by Grant-in-Aid for Young Scientists (B) 15K15941. M. Muramatsu and T. Tsuchiya are supported in part with Grant-in-Aid for Scientific Research (C) 26330025. M. Muramatsu is also partially supported by the Grant-in-Aid for Scientific Research (B)26280005. T. Tsuchiya is also partially supported by the Grant-in-Aid for Scientific Research (B)15H02968.

## Supplementary material

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