Bruno F. Lourenço A convex cone K is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the … Read more
We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly … Read more
We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we … Read more
In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the “eigenvalues” of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret “eigenvalues” in a broad fashion going beyond the usual eigenvalues of matrices. This … Read more
Let \(({\bf P},{\bf D})\) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, \({\bf P}\) and \({\bf D}\) are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function \(v\) as a function of the perturbation is not well-defined at zero … Read more
A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of … Read more