Critical Values of Multilinear Forms under Conic Constraints

Let $\Phi:\Pi_{i=1}^r E_i\to \mathbb{R}$ be a multilinear form on the Cartesian product of finitely many Euclidean vector spaces. We suppose that each factor $E_i$ is equipped with its own closed convex cone $K_i$. We analyze the concept of critical point and critical value of $\Phi$ when each argument of this function is restricted to a normalization constraint and a conic constraint. Our study encompasses the theory of cone-constrained singular values of bilinear and trilinear forms.