Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory

Abstract

Abstract. There are three related concepts that arise in connection with the angular analysis of a convex cone: antipodality, criticality, and Nash equilibria. These concepts are geometric in nature but they can also be approached from the perspective of optimization theory. A detailed angular analysis of polyhedral convex cones has been carried out in a recent work of ours. This note focus on two important classes of non-polyhedral convex cones: elliptic cones in an Euclidean vector space and spectral cones in a space of symmetric matrices.Mathematical subject classification: 52A40, 90C26.

A Variational Approach to Copositive Matrices

“…Formula (6.12) is known or ought to be known. A sketch of the proof runs as follows: First of all, observe that P n is unitarily invariant in the sense that A ∈ P n =⇒ U T AU ∈ P n for all U ∈ O n , with O n denoting the group of orthogonal matrices of order n. By relying on the commutation principle [62,Lemma 4] for variational problems with unitarily invariant data, one obtains the reduction formula . The choice of the ordering mechanism is not essential because R n + is permutation invariant.…”

Inradius and Circumradius of Various Convex Cones Arising in Applications

“…Three dimensional elliptic cones have applications in mechanics [13,47], electromagnetic scattering [46], and many other areas. General background on higher dimensional elliptic cones can be found in [24][25][26]28]. Stern and Wolkowicz [42,43] work with a wider class of elliptic cones, namely, those that are represented as the image of the n -dimensional Lorentz cone under a nonsingular linear transformation.…”

“…[32,33]), but we focus the attention on convex cones. A list of examples of spectral convex cones is provided in [26]. What makes a spectral convex cone K so attractive is that everything boils down to working with the corresponding permutation invariant convex cone Q K .…”

“…, n − 1}, consider the positively homogeneous concave function A ∈ S n → g p (A) = sum of the p smallest eigenvalues of A and the corresponding closed convex cone K p,n = {A ∈ S n : g p (A) ≥ 0}. This cone has been studied under different angles by many authors [1,26,34]. It is known that K p,n is a spectral cone with (32) as associated permutation invariant cone.…”

Critical angles in random polyhedral cones

“…Angular spectra of deterministic convex cones is a theme that has been developed in recent years by Iusem and Seeger [9,[11][12][13][14]. In this work we incorporate randomness into the modeling process.…”