Noticias en español
On extensions of Kenderov’s single-valuedness result for monotone maps
Abstract: One of the most famous single-valuedness results for set-valued maps is due to Kenderov and states that a monotone set-valued operator is single-valued at any point where it is lower semi-continuous. This has been extended in Christensen-Kenderov to monotone maps satisfying a so-called property. Our aim in this work is twofold: First, to prove that the property assumption can be weakened; second to emphasize that these classical single-valuedness results for monotone operators can be obtained, in very simple way, as direct consequences of counterpart results proved for...
Read MoreOptimal shape of an open mining pit
Abstract: Given a vein of ore in the ground, which is to be extracted by open-pit mining, what is the shape which will maximize profit? This problem is often treated by discretization, but we present a new way to handle directly the continuous case. We set it up as an optimal transportation problem, which we solve using Kantorovitch duality. This is joint work with Maurice Queyranne
Read MoreConvexity and Duality in the Economics of Financial Equilibrium
Abstract In classical models of economic equilibrium with markets of goods, agents maximize quasi-concave utility subject to budget constraints based on the value of their initial holdings. When financial markets are added, there are serious complications because of uncertainties over future states which the model must reflect. In that context, the existence of prices that bring supply and demand into balance is far from assured, and current theory is unsatisfying. However, strong results about the existence of equilibrium have recently been obtained by requiring utility functions to...
Read MoreEPI-SPLINES AND EXPONENTIAL EPI-SPLINES: PLIABLE APPROXIMATION TOOLS
Abstract: Approximation theory for functions was, at the outset, mostly concerned with finding best approximating functions that can be (totally) described by a finite number of parameters. This question took another dimension when the information about the function was limited, say its value at some points, but also included some knowledge of its global properties, typically smoothness level, which should be replicated in the approximating one. This gave rise to the theory of splines and its manifold implementations. But in an evolving range of applications, the function is only defined...
Read MoreNewton’s method for solving inclusions using set-valued approximations
Abstract Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super-)linear convergence of the Newton-type iterative process for solving generalized equations. The possibility to choose set-valued approximations allows us to describe several iterative schemes in a unified way (such as inexact Newton method, non-smooth Newton method for semi-smooth functions, inexact proximal point algorithm, etc.). Moreover, it also covers a forward-backward splitting algorithm for finding a common zero of the sum...
Read MoreConvergence of descent methods in non-smooth non-convex optimization
Résumé: Following recent advances, we present a large class of descent methods which converges strongly to a critical point of a non-smooth non-convex function, under the condition that it has a ‘good’ behavior around its critical points. More exactly we consider proper lower semi-continuous functions satisfying the so-called Kurdyka-Lojasiewicz inequality. It is satisfied in finite dimensions by tame functions (like semi-algebraic or bounded sub-analytic functions), while in infinite dimensions we know some useful examples in PDE, for instance energies of elliptic PDE with analytic...
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