João Gouveia In this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of Yannakakis that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave … Read more
This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^(1/4) improving on previous lower bounds. For polygons with v vertices, we show that psd rank … Read more
In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verifi cation problem whose complexity is unknown. Citation April 2013 Article Download View Which Nonnegative Matrices … Read more
We apply theta body relaxations to the $K_i$ cover problem and use this to show polynomial time solvability for certain classes of graphs. In particular, we give an effective relaxation where all $K_i$-$p$-hole facets are valid, addressing an open question of Conforti et al \cite{conforti}. For the triangle free problem, we show for $K_n$ that … Read more
The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, … Read more
In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or ‘lift’ of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over … Read more
We investigate the relationships between various sum of squares (SOS) and semidefinite programming (SDP) relaxations for the sensor network localization problem. In particular, we show that Biswas and Ye’s SDP relaxation is equivalent to the degree one SOS relaxation of Kim et al. We also show that Nie’s sparse-SOS relaxation is stronger than the edge-based … Read more
The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite … Read more
Inspired by a question of Lov\’asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov{\’a}sz’s theta body of the graph. … Read more