The algebraic degree of sparse polynomial optimization

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.