  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.
