Computable representations for convex hulls of low-dimensional quadratic forms

• Authors: Kurt M. Anstreicher;Samuel Burer
• Venue: Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
• Year: 2010

Let $${\mathcal{C}}$$ be the convex hull of points $${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$$. Representing or approximating $${\mathcal{C}}$$is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and $${\mathcal{F}}$$is a simplex, then $${\mathcal{C}}$$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 $${\mathcal{F}}$$is a box, then $${\mathcal{C}}$$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 $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$when $${\mathcal{F}\subset\Re^2}$$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 $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$. When n = 3 and $${\mathcal{F}}$$is a box, we show that a representation for $${\mathcal{C}}$$can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.