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De nombreux systèmes physiques, mécaniques, financiers et
économiques peuvent être décrits par des modèles mathématiques
qui visent à optimiser des fonctions, trouver des équilibres
et effectuer des arbitrages. Souvent, la convexité des ensembles
et des fonctions ainsi que les conditions de monotonie sur les
systèmes d'inéquations qui régissent ces systèmes se
présentent naturellement dans les modèles. C'est dans cet esprit
que nous avons conçu ce livre en mettant l'accent sur une approche
géométrique qui privilégie l'intuition par rapport à une
approche plus analytique. Les démonstrations des résultats
classiques ont été revues dans cette optique et simplifiées. De
nombreux exemples d'applications sont étudiés et des exercices
sont proposés. Ce livre s'adresse aux étudiants en master de
mathématiques appliquées, ainsi qu'aux doctorants, chercheurs et
ingénieurs souhaitant comprendre les fondements de l'analyse
convexe et de la théorie des inéquations variationnelles
monotones.
A function is convex if its epigraph is convex. This geometrical
structure has very strong implications in terms of continuity and
differentiability. Separation theorems lead to optimality
conditions and duality for convex problems. A function is
quasiconvex if its lower level sets are convex. Here again, the geo
metrical structure of the level sets implies some continuity and
differentiability properties for quasiconvex functions. Optimality
conditions and duality can be derived for optimization problems
involving such functions as well. Over a period of about fifty
years, quasiconvex and other generalized convex functions have been
considered in a variety of fields including economies, man agement
science, engineering, probability and applied sciences in
accordance with the need of particular applications. During the
last twenty-five years, an increase of research activities in this
field has been witnessed. More recently generalized monotonicity of
maps has been studied. It relates to generalized convexity off
unctions as monotonicity relates to convexity. Generalized
monotonicity plays a role in variational inequality problems,
complementarity problems and more generally, in equilibrium prob
lems."
A function is convex if its epigraph is convex. This geometrical
structure has very strong implications in terms of continuity and
differentiability. Separation theorems lead to optimality
conditions and duality for convex problems. A function is
quasiconvex if its lower level sets are convex. Here again, the geo
metrical structure of the level sets implies some continuity and
differentiability properties for quasiconvex functions. Optimality
conditions and duality can be derived for optimization problems
involving such functions as well. Over a period of about fifty
years, quasiconvex and other generalized convex functions have been
considered in a variety of fields including economies, man agement
science, engineering, probability and applied sciences in
accordance with the need of particular applications. During the
last twenty-five years, an increase of research activities in this
field has been witnessed. More recently generalized monotonicity of
maps has been studied. It relates to generalized convexity off
unctions as monotonicity relates to convexity. Generalized
monotonicity plays a role in variational inequality problems,
complementarity problems and more generally, in equilibrium prob
lems."
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