Generalized Convexity And Generalized Monotonicity
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Generalized Convexity, Generalized Monotonicity: Recent Results
Author: Jean-Pierre Crouzeix
language: en
Publisher: Springer Science & Business Media
Release Date: 1998-08-31
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.
Generalized Convexity, Generalized Monotonicity and Applications
Author: Andrew Eberhard
language: en
Publisher: Springer Science & Business Media
Release Date: 2005
This volume contains a collection of refereed articles on generalized convexity and generalized monotonicity. The first part of the book contains invited papers by leading experts (J.M. Borwein, R.E. Burkard, B.S. Mordukhovich and H. Tuy) with applications of (generalized) convexity to such diverse fields as algebraic dynamics of the Gamma function values, discrete optimization, Lipschitzian stability of parametric constraint systems, and monotonicity of functions. The second part contains contributions presenting the latest developments in generalized convexity and generalized monotonicity: its connections with discrete and with continuous optimization, multiobjective optimization, fractional programming, nonsmooth Aanalysis, variational inequalities, and its applications to concrete problems such as finding equilibrium prices in mathematical economics, or hydrothermal scheduling. Audience This volume is suitable for faculty, graduate students, and researchers in mathematical programming, operations research, convex analysis, nonsmooth analysis, game theory and mathematical economics.
Handbook of Generalized Convexity and Generalized Monotonicity
Author: Nicolas Hadjisavvas
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-01-16
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.