We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of T, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of T in terms of Lispchitz continuity of [T + ρ-11]-1 and a sufficient condition related to expansivity of T. We also establish surjectivity results for maximal ρ-semimonotone operators.

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.

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.

We present an algorithm for the constrained saddle point problem with a convex-concave function L and convex sets with nonempty interior. The method consists of moving away from the current iterate by choosing certain perturbed vectors. The values of gradients of L at these vectors provide an appropriate direction. Bregman functions allow us to define a curve which starts at the current iterate with this direction, and is fully contained in the interior of the feasible set. The next iterate is obtained by moving along such a curve with a certain step size. We establish convergence to a solution with minimal conditions upon the function L, weaker than Lipschitz continuity of the gradient of L, for instance, and including cases where the solution needs not be unique. We also consider the case in which the perturbed vectors are on certain specific curves starting at the current iterate, in which case another convergence proof is provided. In the case of linear programming, we obtain a family of interior point methods where all the iterates and perturbed vectors are computed with very simple formulae, without factorization of matrices or solution of linear systems, which makes the method attractive for very large and sparse matrices. The method may be of interest for massively parallel computing. Numerical examples for the linear programming case are given.