# Multilinear programming over the simplex

Let $$\triangle_3 \in \mathbb{R}^3$$ be the $$3$$-simplex. I am solving a series of multilinear programming problems that looks like this:

$$\text{Maximize}\sum_{0\leq i, j, k \leq 3} A_{i,j,k} x_i x_j x_k$$

Subject to $$(x_1, x_2, x_3)\in \triangle_3$$, $$x_i \in [a_i, b_i]$$, $$x_0=1$$, and there is no quadratic terms or cubed terms (i.e the objective is linear when you fix the other variables).

I am noticing that all the optimal solutions are vertices of $$\cap_i [a_i, b_i] \cap \triangle_3$$. This suggests the function is convex, but it seems unlikely since multilinear optimisation is usually not convex/concave. And even then, I don't see how to verify that: The Hessian would be a function of at least one of the variables, so it is not straightforward to verify that the Hessian is positive semidefinite.

Any idea how I might be able to prove this property here? This would be extremely helpful and would speed up the optimisation process tremendously.