96 publications from this institution
In this paper we introduce and study a subdifferential that is related to the quasi- convex functions, much as the Fenchel--Moreau subdifferential is related to the convex ones. It is defined for any lower semicontinuous function, through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets. We show that this "quasiconvex" subdifferential is always a cyclically quasimonotone operator that coincides with the Fenchel--Moreau subdifferential whenever the function is convex, and that under mild assumptions, the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function. We also show that the "quasiconvex" subdifferential of a lower semicontinuous function contains the derivatives of its differentiable quasiaffine supports. As a consequence, it contains the subdifferential introduced by Martinez-Legaz and Sach in a recent paper [J. Convex Anal., 6 (1999), pp. 1--12]. Several other properties and calculus rules are also established.
It is a great pleasure for us to praise Rosalind Elster’s long-standing professional and courageous commitment to improve the journal Optimization to a high-level international journal. Our journal...
In this paper we introduce a new approach to representing both TU-games and NTU-games as special economic structures.Instead of representing a game as a market -an exchange economy with concave utility functionsas in the extant literature, we represent an arbitrary game as a coalition production economy with a public good.The economy provides an indirect description of the game.Our model uses the idea of a social planner or arbitrager who, through arbitrage on productive activities, seeks to minimize the net cost of providing the public good subject to the constraint that the society satises its reservation welfare level.Note that in constrast to the prior literature on representing games as economic structures, we require neither that the game be balanced nor that there are large numbers of players.Our approach to representing a game as an economic structure uses techniques from consumer demand theory and, in particular, the notion of a compensated demand correspondence.The main results of this paper exhibit a relationship between cooperative game theory and consumer demand theory.
This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems.
International audience
International audience
This article deals with systems of infinitely many inequalities involving functions that are positively homogeneous over a nonempty convex cone of the Euclidean space. Generalized convex conjugation theory is applied to derive a Farkas-type and a Gale-type theorem for this kind of systems. These results are particularized for linear and min-type inequality systems.
We present a generalization of the strong Fitzpatrick inequality in the context of reflexive Banach spaces, involving a twisted bigger conjugate function. We also introduce a related family of gap functions for maximal monotone inclusion problems.
