Mathematical Programming

, Volume 124, Issue 1–2, pp 33–43 | Cite as

Computable representations for convex hulls of low-dimensional quadratic forms

Full Length Paper Series B

Abstract

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.

Keywords

Quadratic form Convex hull Convex envelope Global optimization Semidefinite programming 

Mathematics Subject Classification (2000)

90C20 90C22 90C26 

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Copyright information

© Springer and Mathematical Programming Society 2010

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA

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