Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the … Read more

On the computational complexity of gap-free duals for semidefinite programming

We consider the complexity of gap-free duals in semidefinite programming. Using the theory of homogeneous cones we provide a new representation of Ramana’s gap-free dual and show that the new formulation has a much better complexity than originally proved by Ramana. Citation COR@L Technical Report, Lehigh University Article Download View On the computational complexity of … Read more

Exact duality for optimization over symmetric cones

We present a strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification. We show important complexity implications of the result to semidefinite and second order conic optimization. The result is an application of Borwein and Wolkowicz’s facial reduction procedure to express the minimal cone. We use Pataki’s simplified analysis and … Read more

Invariance and efficiency of convex representations

We consider two notions for the representations of convex cones: $G$-representation and lifted-$G$-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones … Read more

Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers

We study homogeneous convex cones. We first characterize the extreme rays of such cones in the context of their primal construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results, we prove that every homogeneous cone is facially exposed. We provide an alternative proof of … Read more

Relating Homogeneous Cones and Positive Definite Cones via hBcalgebras

$T$-algebras are non-associative algebras defined by Vinberg in the early 1960’s for the purpose of studying homogeneous cones. Vinberg defined a cone $K(\mathcal A)$ for each $T$-algebra $\mathcal A$ and proved that every homogeneous cone is isomorphic to one such $K(\mathcal A)$. We relate each $T$-algebra $\mathcal A$ with a space of linear operators in … Read more