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Analyse convexe et quasi-convexe ; applications en optimisation

Abstract : This document is a research contribution on Convex Analysis, on Generalized Convexity and on their applications in Optimization Theory. The first part deals with several fundamental questions concerning continuity, differentiability and criteria of coincidence for the class of convex functions. Convexification processes for lower semicontinuous functions are also studied. For the class of quasiconvex functions two approaches are used: an analytic approach, in the spirit of non-smooth analysis, and a geometric one, based on the notion of normal cones to sublevel sets. The second part is devoted to applications to the integration of multivalued operators, to Variational Inequality Problems and to finite and infinite dimensional multicriteria optimization problems. Among the concepts that are introduced for the first time in this work are: the notion of strong cyclic monotonicity, which characterizes the subdifferential of a convex function with a continuous restriction on its domain; the notion of cyclic quasimonotonicity, an intrinsic property of the subdifferentials of quasiconvex functions with important applications in Mathematical Economics; and the notion of proper quasimonotonicity, which characterizes the class of operators for which the associated Minty Variational Inequality problem has at least one solution on every nonempty convex and weakly compact subset of their domains. Let us finally mention a new characterization of the Radon-Nikodym property, and an extension to infinite dimensions of a result of Janin concerning the integration of the class of maximal cyclically submonotone operators, which generalizes a classical result of Rockafellar for maximal cyclically monotone operators.
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Habilitation à diriger des recherches
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https://tel.archives-ouvertes.fr/tel-00001355
Contributor : Aris Daniilidis <>
Submitted on : Friday, May 17, 2002 - 10:01:41 PM
Last modification on : Thursday, March 5, 2020 - 7:14:20 PM
Long-term archiving on: : Monday, September 6, 2010 - 11:26:11 AM

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Aris Daniilidis. Analyse convexe et quasi-convexe ; applications en optimisation. Mathématiques [math]. Université de Pau et des Pays de l'Adour, 2002. ⟨tel-00001355⟩

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