Convexity And Duality In Optimization
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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.
Conjugate Duality And Optimization
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Author : R. Tyrrell Rockafellar
language : en
Publisher: SIAM
Release Date : 1974-01-01
Conjugate Duality And Optimization written by R. Tyrrell Rockafellar and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974-01-01 with Technology & Engineering categories.
The theory of duality in problems of optimization is developed in a setting of finite and infinite dimensional spaces using convex analysis. Applications to convex and nonconvex problems. Expository account containing many new results. (Author).
Convexity And Duality In Optimization
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Author : Jacob Ponstein
language : en
Publisher:
Release Date : 1985-10-01
Convexity And Duality In Optimization written by Jacob Ponstein and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985-10-01 with categories.
Duality In Optimization And Variational Inequalities
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Author : C.j. Goh
language : en
Publisher: Taylor & Francis
Release Date : 2002-05-10
Duality In Optimization And Variational Inequalities written by C.j. Goh and has been published by Taylor & Francis this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-05-10 with Mathematics categories.
This comprehensive volume covers a wide range of duality topics ranging from simple ideas in network flows to complex issues in non-convex optimization and multicriteria problems. In addition, it examines duality in the context of variational inequalities and vector variational inequalities, as generalizations to optimization. Duality in Optimization and Variational Inequalities is intended for researchers and practitioners of optimization with the aim of enhancing their understanding of duality. It provides a wider appreciation of optimality conditions in various scenarios and under different assumptions. It will enable the reader to use duality to devise more effective computational methods, and to aid more meaningful interpretation of optimization and variational inequality problems.
Vector Optimization And Monotone Operators Via Convex Duality
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Author : Sorin-Mihai Grad
language : en
Publisher: Springer
Release Date : 2014-09-03
Vector Optimization And Monotone Operators Via Convex Duality written by Sorin-Mihai Grad and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-03 with Business & Economics categories.
This book investigates several duality approaches for vector optimization problems, while also comparing them. Special attention is paid to duality for linear vector optimization problems, for which a vector dual that avoids the shortcomings of the classical ones is proposed. Moreover, the book addresses different efficiency concepts for vector optimization problems. Among the problems that appear when the framework is generalized by considering set-valued functions, an increasing interest is generated by those involving monotone operators, especially now that new methods for approaching them by means of convex analysis have been developed. Following this path, the book provides several results on different properties of sums of monotone operators.
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.
Conjugate Duality In Convex Optimization
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Author : Radu Ioan-Bot
language : en
Publisher: Springer
Release Date : 2011-03-03
Conjugate Duality In Convex Optimization written by Radu Ioan-Bot and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-03-03 with Business & Economics categories.
The results presented in this book originate from the last decade research work of the author in the ?eld of duality theory in convex optimization. The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary and suf?cient optimality conditions and, consequently, in generatingdifferent algorithmic approachesfor solving mathematical programming problems. The investigations made in this work prove the importance of the duality theory beyond these aspects and emphasize its strong connections with different topics in convex analysis, nonlinear analysis, functional analysis and in the theory of monotone operators. The ?rst part of the book brings to the attention of the reader the perturbation approach as a fundamental tool for developing the so-called conjugate duality t- ory. The classical Lagrange and Fenchel duality approaches are particular instances of this general concept. More than that, the generalized interior point regularity conditions stated in the past for the two mentioned situations turn out to be p- ticularizations of the ones given in this general setting. In our investigations, the perturbationapproachrepresentsthestartingpointforderivingnewdualityconcepts for several classes of convex optimization problems. Moreover, via this approach, generalized Moreau–Rockafellar formulae are provided and, in connection with them, a new class of regularity conditions, called closedness-type conditions, for both stable strong duality and strong duality is introduced. By stable strong duality we understand the situation in which strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional.
Convex Duality And Financial Mathematics
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Author : Peter Carr
language : en
Publisher: Springer
Release Date : 2018-07-18
Convex Duality And Financial Mathematics written by Peter Carr and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-18 with Mathematics categories.
This book provides a concise introduction to convex duality in financial mathematics. Convex duality plays an essential role in dealing with financial problems and involves maximizing concave utility functions and minimizing convex risk measures. Recently, convex and generalized convex dualities have shown to be crucial in the process of the dynamic hedging of contingent claims. Common underlying principles and connections between different perspectives are developed; results are illustrated through graphs and explained heuristically. This book can be used as a reference and is aimed toward graduate students, researchers and practitioners in mathematics, finance, economics, and optimization. Topics include: Markowitz portfolio theory, growth portfolio theory, fundamental theorem of asset pricing emphasizing the duality between utility optimization and pricing by martingale measures, risk measures and its dual representation, hedging and super-hedging and its relationship with linear programming duality and the duality relationship in dynamic hedging of contingent claims
Convex Analysis And Optimization
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Author : Dimitri Bertsekas
language : en
Publisher: Athena Scientific
Release Date : 2003-03-01
Convex Analysis And Optimization written by Dimitri Bertsekas and has been published by Athena Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-03-01 with Mathematics categories.
A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html
Duality In Vector Optimization
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Author : Radu Ioan Bot
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-08-12
Duality In Vector Optimization written by Radu Ioan Bot 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 2009-08-12 with Mathematics categories.
This book presents fundamentals and comprehensive results regarding duality for scalar, vector and set-valued optimization problems in a general setting. One chapter is exclusively consecrated to the scalar and vector Wolfe and Mond-Weir duality schemes.