Recently (1995), Polyak and Kogan have introduced the notion of a principal point for characterizing the boundary of the image of an m-dimensional box Q under a multilinear mapping f : R-m --> C, where R and C denote the real line and the complex plane, respectively. They have also shown that the image of the set of principal points P subset of Q includes the boundary of the image f(Q). However, an efficient and systematic algorithm for finding the set of principal points (PPs) P is still lacking. In this paper, we introduce the set of generalized principal points (GPPs) G, which is a superset of P, to construct the value set f(Q). The GPP set G possesses the following useful properties which facilitate construction of the boundary of the value set f(Q): (i) it has a simpler analytic description than the set of principal points P; (ii) it consists of all edges of the box Q and one-dimensional smooth manifolds on the faces and in the interior of Q; (iii) all GPP manifolds have a connection with at least one edge of the box Q. On the basis of the connectedness property of GPP manifolds, we present a multi-dimensional pivoting procedure with integer labelling to trace out all GPP manifolds. As an application, the presented value-set construction algorithm is applied, along with the zero-inclusion principle and a two-dimensional pivoting procedure, to characterize the smallest set of regions in the complex plane within which all the roots of a multilinear interval polynomial family lie.