From Convexity To Nonconvexity
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From Convexity To Nonconvexity
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Author : R.P. Gilbert
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
From Convexity To Nonconvexity written by R.P. Gilbert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.
This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the main editor who had collected all of the materials, for this volume, P. D. Panagiotopoulos, died unexpectedly during the period when we were editing the manuscript. The other two editors in appreciation of Panagiotopoulos' contribution to this field, believe it is therefore fitting that this collection be dedicated to his memory also. The theme of the collection is centered around the seminal research of G. Fichera on the Signorini problem. Variants on this idea enter in different ways. For example, by bringing in friction the problem is no longer self-adjoint and the minimization formulation is not valid. A large portion of this collection is devoted to survey papers concerning hemivariational methods, with a main point of its application to nonsmooth mechanics. Hemivariational inequali ties, which are a generalization of variational inequalities, were pioneered by Panagiotopoulos. There are many applications of this theory to the study of non convex energy functionals occurring in many branches of mechanics. An area of concentration concerns contact problems, in particular, quasistatic and dynamic contact problems with friction and damage. Nonsmooth optimization methods which may be divided into the main groups of subgradient methods and bundle methods are also discussed in this collection.
Duality For Nonconvex Approximation And Optimization
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Author : Ivan Singer
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-03-12
Duality For Nonconvex Approximation And Optimization written by Ivan Singer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-03-12 with Mathematics categories.
The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.
Generalized Convexity Generalized Monotonicity Recent Results
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Author : Jean-Pierre Crouzeix
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
Generalized Convexity Generalized Monotonicity Recent Results written by Jean-Pierre Crouzeix and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.
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.
Abstract Convexity And Global Optimization
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Author : Alexander M. Rubinov
language : en
Publisher: Springer Science & Business Media
Release Date : 2000-05-31
Abstract Convexity And Global Optimization written by Alexander M. Rubinov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-05-31 with Mathematics categories.
This book consists of two parts. Firstly, the main notions of abstract convexity and their applications in the study of some classes of functions and sets are presented. Secondly, both theoretical and numerical aspects of global optimization based on abstract convexity are examined. Most of the book does not require knowledge of advanced mathematics. Classical methods of nonconvex mathematical programming, being based on a local approximation, cannot be used to examine and solve many problems of global optimization, and so there is a clear need to develop special global tools for solving these problems. Some of these tools are based on abstract convexity, that is, on the representation of a function of a rather complicated nature as the upper envelope of a set of fairly simple functions. Audience: The book will be of interest to specialists in global optimization, mathematical programming, and convex analysis, as well as engineers using mathematical tools and optimization techniques and specialists in mathematical modelling.
Handbook Of Generalized Convexity And Generalized Monotonicity
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Author : Nicolas Hadjisavvas
language : en
Publisher:
Release Date : 2005
Handbook Of Generalized Convexity And Generalized Monotonicity written by Nicolas Hadjisavvas and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Convex functions categories.
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.
Abstract Convexity And Global Optimization
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Author : Alexander M. Rubinov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Abstract Convexity And Global Optimization written by Alexander M. Rubinov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.
Convex Analysis And Beyond
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Author : Boris S. Mordukhovich
language : en
Publisher: Springer Nature
Release Date : 2022-04-24
Convex Analysis And Beyond written by Boris S. Mordukhovich and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-24 with Mathematics categories.
This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications. The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.
Advances In Convex Analysis And Global Optimization
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Author : Nicolas Hadjisavvas
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
Advances In Convex Analysis And Global Optimization written by Nicolas Hadjisavvas and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.
There has been much recent progress in global optimization algo rithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. Convex analysis plays a fun damental role in the analysis and development of global optimization algorithms. This is due essentially to the fact that virtually all noncon vex optimization problems can be described using differences of convex functions and differences of convex sets. A conference on Convex Analysis and Global Optimization was held during June 5 -9, 2000 at Pythagorion, Samos, Greece. The conference was honoring the memory of C. Caratheodory (1873-1950) and was en dorsed by the Mathematical Programming Society (MPS) and by the Society for Industrial and Applied Mathematics (SIAM) Activity Group in Optimization. The conference was sponsored by the European Union (through the EPEAEK program), the Department of Mathematics of the Aegean University and the Center for Applied Optimization of the University of Florida, by the General Secretariat of Research and Tech nology of Greece, by the Ministry of Education of Greece, and several local Greek government agencies and companies. This volume contains a selective collection of refereed papers based on invited and contribut ing talks presented at this conference. The two themes of convexity and global optimization pervade this book. The conference provided a forum for researchers working on different aspects of convexity and global opti mization to present their recent discoveries, and to interact with people working on complementary aspects of mathematical programming.
Convexity And Duality In Optimization
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Author : Jacob Ponstein
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Convexity And Duality In Optimization written by Jacob Ponstein and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Business & Economics categories.
The analysis and optimization of convex functions have re ceived a great deal of attention during the last two decades. If we had to choose two key-words from these developments, we would retain the concept of ~ubdi66~e~ and the duality theo~y. As it usual in the development of mathematical theories, people had since tried to extend the known defi nitions and properties to new classes of functions, including the convex ones. For what concerns the generalization of the notion of subdifferential, tremendous achievements have been carried out in the past decade and any rna·· thematician who is faced with a nondifferentiable nonconvex function has now a panoply of generalized subdifferentials or derivatives at his disposal. A lot remains to be done in this area, especially concerning vecto~-valued functions ; however we think the golden age for these researches is behind us. Duality theory has also fascinated many mathematicians since the underlying mathematical framework has been laid down in the context of Convex Analysis. The various duality schemes which have emerged in the re cent years, despite of their mathematical elegance, have not always proved as powerful as expected.
The Theory Of Subgradients And Its Applications To Problems Of Optimization
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Author : R. Tyrrell Rockafellar
language : en
Publisher:
Release Date : 1981
The Theory Of Subgradients And Its Applications To Problems Of Optimization written by R. Tyrrell Rockafellar and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with Mathematics categories.