Author: Olvi L. Mangasarian
Publisher:
ISBN:
Category : Convex functions
Languages : en
Pages : 17
Get Book Here
Book Description
A number of recent results which establish the convexity, pseudo-convexity or quasi-convexity of certain functions are shown to be special cases of the fact that under suitable conditions a composite function is convex, pseudo-convex or quasi-convex. (Author).
Author: Jacques A. Ferland
Publisher:
ISBN:
Category : Functions
Languages : en
Pages : 84
Get Book Here
Book Description
The purpose of the paper is to prove that testing quasi-convexity (pseudo-convexity) of quadratic functions on solid convex sets can be reduced to an examination of finitely many conditions. One determines two maximal domains of quasi-convexity (pseudo-convexity) for the quadratic form Psi(x) = (x, Dx) where D has exactly one negative eigenvalue, and conversely, one shows that if the quadratic form Psi is quasi-convex (pseudo-convex) on a solid convex set, then the matrix D has exactly one negative eignevalue and the solid convex set is contained in one of the maximal domains. The special case when the solid convex set is the nonnegative (semi-positive) orthant is also analyzed. This study is then extended to quadratic functions Phi(x) = 1/2(x, Dx) + (c, x). Analogous results hold under the additional condition that the set (a/Da+c = 0) is not empty. In the last part of this paper, one analyzes functions that are not necessarily quadratic. One obtains some results on mathematical programming problems having twice differentiable quasi-convex objective function and constraint functions. Finally, one gives a necessary condition and a sufficient condition for the quasi-convexity of a function in Class C squared (i.e., twice continuously differentiable) on a solid convex set. One also establishes a relation between the quasi-convexity and the pseudo-convexity of twice differentiable functions on solid convex sets. (Author).
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 0817645853
Category : Mathematics
Languages : en
Pages : 424
Get Book Here
Book Description
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.
Author: Jean-Pierre Crouzeix
Publisher: Springer Science & Business Media
ISBN: 9780792350880
Category : Mathematics
Languages : en
Pages : 496
Get Book Here
Book Description
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.
Author: Nicolas Hadjisavvas
Publisher: Springer Science & Business Media
ISBN: 0387233938
Category : Mathematics
Languages : en
Pages : 684
Get Book Here
Book Description
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.
Author: Constantin P. Niculescu
Publisher: Springer
ISBN: 3319783378
Category : Mathematics
Languages : en
Pages : 430
Get Book Here
Book Description
Thorough introduction to an important area of mathematics Contains recent results Includes many exercises
Author: Jonathan M. Borwein
Publisher: Cambridge University Press
ISBN: 1139811096
Category : Mathematics
Languages : en
Pages : 533
Get Book Here
Book Description
Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.
Author: Mordecai Avriel
Publisher: Courier Corporation
ISBN: 9780486432274
Category : Mathematics
Languages : en
Pages : 548
Get Book Here
Book Description
This overview provides a single-volume treatment of key algorithms and theories. Begins with the derivation of optimality conditions and discussions of convex programming, duality, generalized convexity, and analysis of selected nonlinear programs, and then explores techniques for numerical solutions and unconstrained optimization methods. 1976 edition. Includes 58 figures and 7 tables.
Author: Mordecai Avriel
Publisher: SIAM
ISBN: 0898718961
Category : Mathematics
Languages : en
Pages : 342
Get Book Here
Book Description
Originally published: New York: Plenum Press, 1988.
Author: Jacques A. Ferland
Publisher:
ISBN:
Category :
Languages : en
Pages : 74
Get Book Here
Book Description
