Jean-Pierre Crouzeix - Böcker

The geometrical structure induced by convexity in mathematical programming has many useful properties: continuity and differentiability of the functions, separability and optimality conditions, duality, sensibility of the optimal solutions, and so on. Several of the most interesting ones are preserved when convexity is relaxed in quasiconvexity or pseudoconvexity (a function is quasi-convex if its lower level sets are convex). This is still the case for variational inequalities problems when the classical monotonicity assumption on the map is relaxed in quasimonotonicity or pseudomonotonicity. This volume contains 23 selected lectures presented at an international symposium on generalized convexity. It provides a review of developments. The text should be of value to researchers and students working in economics, mathematical programming, operations research, management sciences, equilibrium problems, engineering and probability.

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.

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.