The algebraic degree of sparse polynomial optimization

Author:

Julia Lindberg, Leonid Monin, Kemal Rose

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Optimization and Control (math.OC)

journal:

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date:

2023-08-14 16:00:00

Abstract

In this paper we study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one as an intersection product on a toric variety. As a corollary, we obtain a convex geometric interpretation of polar degrees, a classical invariant of algebraic varieties as well as Euclidean distance degrees. Furthermore, we prove BKK generality of Lagrange systems in many instances. Motivated by our result expressing the algebraic degree of sparse polynomial optimisation problems via Porteus' formula, in the appendix we answer a related question concerning the degree of sparse determinantal varieties.