On the convexification of a bilinearly constrained set with a common variable

Abstract

We study the convexification of sets defined by box constraints and bilinear inequality constraints in which all bilinear terms share a common variable. For this structured class of nonconvex sets, we provide a complete characterization of the extreme points of the convex hull and derive an explicit description of all facet-defining inequalities, thereby obtaining a full polyhedral description of the convex hull. In addition, we develop a polynomial-time separation algorithm for the facet-defining inequalities and illustrate its effectiveness through examples. These results offer both structural insight into the geometry of the convex hull and algorithmic tools for identifying violated inequalities.