Computable representations for convex hulls of low-dimensional quadratic forms
Let C be the convex hull of points {(1 x) ((1) x)(T) [x is an element of F subset of R(n)}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if 17 <= 4 and F is a simplex, then C has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and F is a box, then C has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of {(x(1), x(2), x(1)x(2)) vertical bar x is an element of F} when F subset of R(2) is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ((x(1), x(2), x(1)x(2)) vertical bar x is an element of F}. When n = 3 and F is a box, we show that a representation for C can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.
convex hull quadratic form global optimization convex envelope semidefinite programming