Alberto Seeger

A Variational Approach to Copositive Matrices

This work surveys essential properties of the so-called copositive matrices, the study of which has been spread over more than fifty-five years. Special emphasis is given to variational aspects related to the concept of copositivity. In addition, some new results on the geometry of the cone of copositive matrices are presented here for the first time.

Cone-constrained eigenvalue problems: theory and algorithms

Equilibria in mechanics or in transportation models are not always expressed through a system of equations, but sometimes they are characterized by means of complementarity conditions involving a convex cone. This work deals with the analysis of cone-constrained eigenvalue problems. We discuss some theoretical issues like, for instance, the estimation of the maximal number of eigenvalues in a cone-constrained problem. Special attention is paid to the Paretian case. As a short addition to the theoretical part, we introduce and study two algorithms for solving numerically such type of eigenvalue problems.

On eigenvalues induced by a cone constraint

Abstract Let A be an n×n real matrix, and K⊂ R n be a closed convex cone. The spectrum of A relative to K, denoted by σ(A,K), is the set of all λ∈ R for which the linear complementarity problem x∈K, Ax−λx∈K + , 〈x,Ax−λx〉=0 admitsa nonzero solution x∈ R n . The notation K+ stands for the (positive) dual cone of K. The purpose of this work is to study the main properties of the mapping σ(·,K), and discuss some structural differences existing between the polyhedral case (i.e. K is finitely generated) and the nonpolyhedral case.

A nonsmooth algorithm for cone-constrained eigenvalue problems

AbstractWe study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form

Axiomatization of the index of pointedness for closed convex cones

K\ni x\perp(Ax-\lambda Bx)\in K^{+}.

Searching for critical angles in a convex cone

Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. The symbol K refers to a closed convex cone in the Euclidean space ℝn and (A,B) is a pair of possibly asymmetric matrices of order n. Special attention is paid to the case in which K is the nonnegative orthant of ℝn. The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.

Second Order Directional Derivatives in Parametric Optimization Problems

Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity system K ∋ x ⊥(Ax-λ Bx) ∈ K+. This problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.

Convex Analysis of Spectrally Defined Matrix Functions

Let C(H) denote the class of closed convex cones in a Hilbert space H. One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. This approach has been explored in detail in a previous work of ours. We now go beyond this particular choice and set up an axiomatic background for addressing this issue. We define an index of pointedness over H as being a function f: C(H) ® R satisfying a certain number of axioms. The number f(K) is intended, of course, to measure the degree of pointedness of the cone K. Although several important examples are discussed to illustrate the theory in action, the emphasis of this work lies in the general properties that can be derived directly from the axiomatic model.