The algebraic degree of sparse polynomial optimization

TL;DR

This paper investigates the algebraic complexity of sparse polynomial optimization problems, providing geometric characterizations of critical points and linking classical invariants to convex geometry, with implications for computational algebraic geometry.

Contribution

It introduces new geometric characterizations of critical points in sparse polynomial optimization and connects algebraic invariants to convex geometric interpretations.

Findings

Critical points characterized as mixed volume and intersection product

Convex geometric interpretation of polar and Euclidean distance degrees

Proof of BKK generality of Lagrange systems in various cases

Abstract

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 the BKK generality of Lagrange systems in many instances.