  Disjointly constrained multilinear programming concerns the problem of
maximizing a multilinear function on the product of finitely many disjoint
polyhedra. While maximizing a linear function on a polytope (linear
programming) is known to be solvable in polynomial time, even bilinear
programming is NP-hard. Based on a reformulation of the problem in terms of
sum-of-squares polynomials, we study a hierarchy of semidefinite relaxations to
the problem. It follows from the general theory that the sequence of optimal
values converges asymptotically to the optimal value of the multilinear
program. We show that the semidefinite hierarchy converges generically in
finitely many steps to the optimal value of the multilinear problem. We outline
two applications of the main result. For nondegenerate bimatrix games, a Nash
equilibrium can be computed by the sum of squares approach in finitely many
steps. Under an additional geometric condition, the NP-complete containment
problem for projections of $\mathcal{H}$-polytopes can be decided in finitely
many steps.
