Abstract
We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.
Due to space limitations, we omit several proofs. These can be found at
https://www.math.ucdavis.edu/~deloera/IPCO2024.pdf.
This research is partially based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the first, third, and fourth authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Discrete Optimization program. We thank Kurt Anstreicher, Renata Sotirov, Pablo Parrilo, Chiara Meroni, and Bento Natura for relevant comments.
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De Loera, J.A., Marsters, B., Xu, L., Zhang, S. (2024). Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham. https://doi.org/10.1007/978-3-031-59835-7_8
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