Various generalizations of the classical concept of a convex function have been introduced, especially during the second half of the 20th century. Generalized convex functions are the many nonconvex functions which share at least one of the valuable properties of convex functions. Apart from their theoretical interest, they are often more suitable than convex functions to describe real-word problems in disciplines such as economics, engineering, management science, probability theory and in other applied sciences. More recently, generalized monotone maps which are closely related to generalized convex functions have also been studied extensively. While initial efforts to generalize convexity and monotonicity were limited to only a few research centers, today there are numerous researchers throughout the world and in various disciplines engaged in theoretical and applied studies of generalized convexity/monotonicity (see http://www.genconv.org).

The Handbook offers a systematic and thorough exposition of the theory and applications of the various aspects of generalized convexity and generalized monotonicity. It is aimed at the non-expert, for whom it provides a detailed introduction, as well as at the expert who seeks to learn about the latest developments and references in his research area. Results in this fast growing field are contained in a large number of scientific papers which appeared in a variety of professional journals, partially due to the interdisciplinary nature of the subject matter. Each of its fourteen chapters is written by leading experts of the respective research area starting from the very basics and moving on to the state of the art of the subject.

Each chapter is complemented by a comprehensive bibliography which will assist the non-expert and expert alike.

Résumé
Studies in generalized convexity and generalized monotonicity have significantly increased during the last two decades. Researchers with very diverse backgrounds such as mathematical programming, optimization theory, convex analysis, nonlinear analysis, nonsmooth analysis, linear algebra, probability theory, variational inequalities, game theory, economic theory, engineering, management science, equilibrium analysis, for example are attracted to this fast growing field of study. Such enormous research activity is partially due to the discovery of a rich, elegant and deep theory which provides a basis for interesting existing and potential applications in different disciplines. The handbook offers an advanced and broad overview of the current state of the field. It contains fourteen chapters written by the leading experts on the respective subject; eight on generalized convexity and the remaining six on generalized monotonicity.

Contenu

to Convex and Quasiconvex Analysis.- Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case.- Continuity and Differentiability of Quasiconvex Functions.- Generalized Convexity and Optimality Conditions in Scalar and Vector Optimization.- Generalized Convexity in Vector Optimization.- Generalized Convex Duality and its Economic Applicatons.- Abstract Convexity.- Fractional Programming.- Generalized Monotone Maps.- Generalized Convexity and Generalized Derivatives.- Generalized Convexity, Generalized Monotonicity and Nonsmooth Analysis.- Pseudomonotone Complementarity Problems and Variational Inequalities.- Generalized Monotone Equilibrium Problems and Variational Inequalities.- Uses of Generalized Convexity and Generalized Monotonicity in Economics.