On the spherical convexity of quadratic functions

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cones are given

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras

Let L be a linear transformation on a finite dimensional real Hilbert space H and K be a closed convex cone with dual K ∗ in H . The cone spectrum of L relative to K is the set of all real λ for which the linear complementarity problem x ∈ K , y = L ( x ) - λ x ∈ K ∗ , and 〈 x , y 〉 = 0 admits a nonzero solution x . In the setting of a Euclidean Jordan algebra H and the corresponding symmetric cone K , we discuss the finiteness of the cone spectrum for Z -transformations and quadratic representations on H

Convexity of sets and quadratic functions on the hyperbolic space

In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first and second order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented