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.

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Complementarity conditions, Generalized eigenvalue problems, Convex cones,

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Abstract. 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. 47L07, 52A20. Mathematical subject classification:

International audienceWe study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form Kx(Ax−Bx)K+ 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 ℝ n 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 ℝ n . The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail

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Abstract. 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 systemThis 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.Mathematical subject classification: 65F15, 90C33.