Noticias en español
Satisfying Instead of Optimizing in the Nash Demand Games.
Abstract: The Nash Demand Game (NDG) has been one of the first models (Nash 1953) that has tried to describe the process of negotiation, competition, and cooperation. This model is still subject to active research, in fact, it maintains a set of open questions regarding how agents optimally select their decisions and how they face uncertainty. However, the agents act rather guided by chance and necessity, with a Darwinian flavor. Satisfying, instead of optimising. The Viability Theory (VT) has this approach. Therefore, we investigate the NDG under this point of view. In...
Read MoreEnlargements of the Moreau-Rockafellar Subdifferential.
Abstract: The Moreau–Rockafellar subdifferential is a highly important notion in convex analysis and optimization theory. But there are many functions which fail to be subdifferentiable at certain points. In particular, there is a continuous convex function defined on $\ell^2(\mathbb{N})$, whose Moreau–Rockafellar subdifferential is empty at every point of its domain. This talk proposes some enlargements of the Moreau—Rockafellar subdifferential: the sup$^\star$-sub\-differential, sup-subdifferential and symmetric subdifferential, all of them being nonempty for the...
Read MoreA general asymptotic function with applications
Abstract: Due to its definition through the epigraph, the usual asymptotic function of convex analysis is a very effective tool for studying minimization, especially of a convex function. However, it is not as convenient, if one wants to study maximization of a function “f”; this is done usually through the hypograph or, equivalently, through “−f”. We introduce a new concept of asymptotic function which allows us to simultaneously study convex and concave functions as well as quasi-convex and quasi-concave functions. We provide some calculus rules and relevant...
Read MoreCharacterizing the calmness property in convex semi-infinite optimization. Modulus estimates
Abstract: We present an overview of the main results on calmness in convex semi-infinite optimization. The first part addresses the calmness of the feasible set and the optimal set mappings for the linear semi-infinite optimization problem in the setting of canonical perturbations, and also in the framework of full perturbations. While there exists a clear proportionality between the calmness moduli of the feasible set mappings in both contexts, the analysis of the relationship between the calmness moduli of the argmin mappings is much more complicated. Point-based expressions (only...
Read MoreA VARIATIONAL APPROACH TO SECOND-ORDER OPTIMALITY
ABSTRATC: Conditions associated with local optimality, whether necessary or sufficient, have traditionally been approached through techniques of generalized differentiation. On the first-order level, this has been a long-standing success, although serious challenges remain for equilibrium constraints and the like. On the second-order level, a difficulty areses with the complex concepts of generalized second derivatives and the sometimes-inadequate calculus for determining them. In fact, sufficient second-order conditions of a practical sort, which are the most important aid for numerical...
Read MoreTheorems of Borsuk-Ulam Type
Abstract: The Borsuk-Ulam Theorem states that for any continuous function f from S^n to R^n there is some x in S^n such that f(x) = f(-x). Replace S^n by the boundary of some open set A of E=R^{n+1} and replace R^n by some n dimensional manifold B. The conclusion of the theorem remains, with the pair x, -x replaced by some x,y on the boundary whose convex combinations contain some fixed point z in the interior of that open set. Indeed there is a topological structure to all such solutions when the z is considered a variable. If B is not a manifold, the conclusion fails. However if we allow...
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