Non-exposed faces of the cone of completely positive matrices
Abstract
In this paper, we consider the cone of completely positive matrices. Currently, some families of non-exposed polyhedral faces of this cone were constructed. Inspired by these results, in this paper, we continue the study of the existence and properties of non-exposed faces of the cone of completely positive matrices. We prove a criterion for a face of this cone to be non-exposed. We also provide sufficient conditions that can be easily checked numerically. We show that for any $p\geqslant 6$, there exist non-exposed non-polyhedral faces of the cone of $p\times p$ completely positive matrices. Illustrative examples are given
Keywords
conic optimization,
completely positive matrices,
$K$-semidefinite matrices,
a face of a cone,
exposed and non-exposed faces of a cone
Supporting agencies: This work was supported by the Institute of Mathematics of the National Academy of Sciences of Belarus within the framework of the state programme ”Convergence–2025”, tasks 1.3.01 and 1.3.04.
About the Author
O. I. Kostyukova
Institute of Mathematics of the National Academy of Sciences of Belarus,
Belarus
Belarus
Minsk
References
1. Anjos M. F., Lasserre J. B. (eds). Handbook on Semi-definite, Conic and Polynomial Optimization, International Series in OR/MS, 2012, vol. 166, Springer, New York. https://doi.org/10.1007/978-1-4614-0769-0
2. Letchford A. N., Parkes A. J. A guide to conic optimisation and its applications. RAIRO – Oper. Res., 2018, vol. 52, iss. 4–5, pp. 1087–1106. https://doi.org/10.1051/ro/2018034
3. Bomze I. M., Schachinger W., Uchida G. Think co(mpletely)positive! Matrix properties, examples and a clustered bibliography on copositive optimization. Journal of Global Optimization, 2012, vol. 52, iss. 3, pp. 423–445. https://doi.org/10.1007/s10898-011-9749-3
4. Dickinson P. J. Geometry of the copositive and completely positive cones. Journal of Mathematical Analysis and Applications, 2011, vol. 380, iss. 1, pp. 377–395. https://doi.org/10.1016/j.jmaa.2011.03.005
5. Kostyukova O. I., Tchemisova T. V. On equivalent representations and properties of faces of the cone of copositive matrices. Optimization, 2022, vol. 71, iss. 11, pp. 3211–3239. https://doi.org/10.1080/02331934.2022.2027939
6. Hoffman A. J., Pereira F. On copositive matrices with -1, 0, 1 entries. Journal of Combinatorial Theory, 1973, Series A, vol. 14, iss. 3, pp. 302–309. https://doi.org/10.1016/0097-3165(73)90006-X
7. Berman A., Du¨ r M., Shaked-Monderer N. Open problems in the theory of completely positive and copositive matrices. The Electronic Journal of Linear Algebra, 2015, vol. 29, pp. 46– 58. https://doi.org/10.13001/1081-3810.2943
8. Zhang Q. Completely positive cones: are they facially exposed? Linear Algebra and its Applications, 2018, vol. 558, pp. 195–204. https://doi.org/10.1016/j.laa.2018.08.028
9. Zhang Q. Faces of the 5 × 5 completely positive cone. Linear and Multilinear Algebra, 2020, vol. 68, iss. 12, pp. 2523–2540. https://doi.org/10.1080/03081087.2019.1586827
10. Kostyukova O. I. Non-exposed polyhedral faces of the completely positive cone. Linear and Multilinear Algebra, 2024, pp. 1–28. https://doi.org/10.1080/03081087.2024.2346313
11. Eichfelder G., Jahn J. Set-Semidefinite Optimization. Journal of Convex Analysis, 2008, vol. 15, iss. 4, pp. 767–801.
12. Kostyukova O. I., Tchemisova T. V., Dudina O. S. On the Uniform Duality in Copositive Optimization. Journal of Optimization Theory and Applications, 2024. https://doi.org/10.1007/s10957-024-02515-1
13. Wang F., Wolkowicz H. Singularity degree of non-facially exposed faces. 2022, arXiv preprint. arXiv: 2211.00834.
14. Kostyukova O. I., Tchemisova T. V. Representation of Zeros of a Copositive Matrix via Maximal Cliques of a Graph. 2024. arXiv preprint. arXiv: 2410.08066.
15. Alizadeh F., Schmieta S. H. Optimization with Semi-definite, Quadratic and Linear Constraints, Report 23-97. 1997. Rutgers Center for Operations Research, Rutgers University.
Review
For citations:
Kostyukova O.I. Non-exposed faces of the cone of completely positive matrices. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(2):56-68.
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