Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, … Read more
Principal angles between linear subspaces have been studied for their application to statistics, numerical linear algebra, and other areas. In 2005, Iusem and Seeger defined critical angles within a single convex cone as an extension of antipodality in a compact set. Then, in 2016, Seeger and Sossa extended that notion to two cones. This was … Read more
We address the issue of measuring the degree of axial symmetry of a convex cone. By following an axiomatic approach, we introduce and explore the concept of axial symmetry index. This concept is illustrated with the help of several interesting examples. By way of application, we establish a conic version of the Blekherman inequality concerning … Read more
The problem of measuring the degree of central symmetry of a convex body has been treated by various authors since the early twentieth century. This work addresses the issue of measuring the degree of axial symmetry of a convex cone. Passing from central symmetry in convex bodies to axial symmetry in convex cones is not … Read more
By using an axiomatic approach we formalize the concept of size index for closed convex cones in the Euclidean space $\mathbb{R}^n$. We review a dozen of size indices disseminated through the literature, commenting on the advantages and disadvantages of each choice. CitationTo appear in Journal of Convex Analysis (2015) ArticleDownload View PDF
Up to orthogonal transformation, a solid closed convex cone $K$ in the Euclidean space $\mathbb{R}^{n+1}$ is the epigraph of a nonnegative sublinear function $f:\mathbb{R}^n\to \mathbb{R}$. This work explores the link between the geometric properties of $K$ and the analytic properties of $f$. CitationJOURNAL OF CONVEX ANALYSIS, 2011, in press. ArticleDownload View PDF
This is the second part of a work devoted to the theory of epigraphical cones and their applications. A convex cone $K$ in the Euclidean space $\mathbb{R}^{n+1}$ is an epigraphical cone if it can be represented as epigraph of a nonnegative sublinear function $f: \mathbb{R}^n\to \mathbb{R}$. We explore the link between the geometric properties of … Read more
In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As … Read more
There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of epsilon-optimal solutions, and (ii) the maximum distance of epsilon-optimal solutions to the … Read more