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arXiv · 2403.09927

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones

Jesús A. De Loera·Brittney Marsters·Luze Xu·Shixuan Zhang
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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-Chv\'{a}tal closure, and Carath\'{e}odory rank to integer points in arbitrary cones.

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Jesús A. De Loera, Brittney Marsters, Luze Xu, Shixuan Zhang. 2024-03-14. Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones. https://doi.org/10.1007/s10107-024-02188-8

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