Convex Analysis and Global Optimization

Convex Analysis and Global Optimization

This book develops a coherent and rigorous theory of deterministic global optimization from this point of view.

Author: Hoang Tuy

Publisher: Springer Science & Business Media

ISBN: 0792348184

Category: Business & Economics

Page: 362

View: 678

Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development. Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.
Categories: Business & Economics

Convex Analysis and Global Optimization

Convex Analysis and Global Optimization

Global optimization is concerned with finding global solutions to nonconvex optimization problems. Although until recently convex analysis has been developed mainly under the impulse of convex and local optimization, it has become a ...

Author: Hoang Tuy

Publisher: Springer

ISBN: 9783319314846

Category: Mathematics

Page: 505

View: 846

This book presents state-of-the-art results and methodologies in modern global optimization, and has been a staple reference for researchers, engineers, advanced students (also in applied mathematics), and practitioners in various fields of engineering. The second edition has been brought up to date and continues to develop a coherent and rigorous theory of deterministic global optimization, highlighting the essential role of convex analysis. The text has been revised and expanded to meet the needs of research, education, and applications for many years to come. Updates for this new edition include: · Discussion of modern approaches to minimax, fixed point, and equilibrium theorems, and to nonconvex optimization; · Increased focus on dealing more efficiently with ill-posed problems of global optimization, particularly those with hard constraints; · Important discussions of decomposition methods for specially structured problems; · A complete revision of the chapter on nonconvex quadratic programming, in order to encompass the advances made in quadratic optimization since publication of the first edition. · Additionally, this new edition contains entirely new chapters devoted to monotonic optimization, polynomial optimization and optimization under equilibrium constraints, including bilevel programming, multiobjective programming, and optimization with variational inequality constraint. From the reviews of the first edition: The book gives a good review of the topic. ...The text is carefully constructed and well written, the exposition is clear. It leaves a remarkable impression of the concepts, tools and techniques in global optimization. It might also be used as a basis and guideline for lectures on this subject. Students as well as professionals will profitably read and use it.—Mathematical Methods of Operations Research, 49:3 (1999)
Categories: Mathematics

Advances in Convex Analysis and Global Optimization

Advances in Convex Analysis and Global Optimization

Chapter 37 CONVEXITY AND MONOTONICITY IN GLOBAL OPTIMIZATION Hoang Tuy Institute of Mathematics P.O. Boa: 631 htuy(0.hn.vnn.vn Abstract Keywords: , Bo Ho, Hanoi, Vietnam Convexity and monotonicity are two properties of crucial ...

Author: Nicolas Hadjisavvas

Publisher: Springer Science & Business Media

ISBN: 9781461302797

Category: Mathematics

Page: 597

View: 265

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.
Categories: Mathematics

Advances in Convex Analysis and Global Optimization

Advances in Convex Analysis and Global Optimization

This volume contains a selection of refereed papers based on invited and contributing talks presented at the conference. The two themes of convexity and global optimization pervade the book.

Author: Constantin Carathéodory

Publisher: Springer Science & Business Media

ISBN: 0792369424

Category: Computers

Page: 630

View: 449

There has been much recent progress in global optimization algorithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. Convex analysis plays a fundamental role in the analysis and development of global optimization algorithms. This is due to the fact that virtually all nonconvex 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 June 5-9, 2000 at Pythagorian, Samos, Greece. It was in honor of the memory of C. Caratheodory (1873-1950). It was endorsed by the Mathematical Programming Society (MPS) and by the Society for industrial and Applied Mathematics (SIAN) Activity Group in Optimization. This volume contains a selection of refereed papers based on invited and contributing talks presented at the conference. The two themes of convexity and global optimization pervade the book. The conference provided a forum for researchers working on different aspects of convexity and global optimization to present their recent discoveries, and to interact with people working on complementary aspects of mathematical programming. Audience: Faculty, graduate students, and researchers in mathematical programming, computer science, and engineering.
Categories: Computers

Abstract Convexity and Global Optimization

Abstract Convexity and Global Optimization

This book consists of two parts.

Author: Alexander M. Rubinov

Publisher: Springer Science & Business Media

ISBN: 079236323X

Category: Mathematics

Page: 516

View: 164

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.
Categories: Mathematics

Abstract Convexity and Global Optimization

Abstract Convexity and Global Optimization

analysis: each lower semicontinuous convex function f is the upper envelope (the point-wise supremum) of the set of all affine functions which are minored by f. This result is closely related to the existence of global affine supports.

Author: Alexander M. Rubinov

Publisher: Springer Science & Business Media

ISBN: 9781475732009

Category: Mathematics

Page: 493

View: 689

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.
Categories: Mathematics

Advances in Applied Mathematics and Global Optimization

Advances in Applied Mathematics and Global Optimization

Convex analysis has wide applications in science and engineering, such as mechanics, optimization and control, theoretical statistics, mathematical economics and game theory, and so on. It offers an analytic framework to treat systems ...

Author: David Y. Gao

Publisher: Springer Science & Business Media

ISBN: 9780387757148

Category: Mathematics

Page: 520

View: 233

The articles that comprise this distinguished annual volume for the Advances in Mechanics and Mathematics series have been written in honor of Gilbert Strang, a world renowned mathematician and exceptional person. Written by leading experts in complementarity, duality, global optimization, and quantum computations, this collection reveals the beauty of these mathematical disciplines and investigates recent developments in global optimization, nonconvex and nonsmooth analysis, nonlinear programming, theoretical and engineering mechanics, large scale computation, quantum algorithms and computation, and information theory.
Categories: Mathematics

Variational Analysis and Set Optimization

Variational Analysis and Set Optimization

Journal of Global Optimization, 45: 355–369, 2009. [8] J. Dutta, J. E. Martinez-Legaz and A. M. Rubinov. Monotonic analysis over cones: III. Journal of Convex Analysis, 15:561–579, 2008. [9] V. Jeyakumar. Characterizing set containments ...

Author: Akhtar A. Khan

Publisher: CRC Press

ISBN: 9781351712064

Category: Business & Economics

Page: 226

View: 728

This book contains the latest advances in variational analysis and set / vector optimization, including uncertain optimization, optimal control and bilevel optimization. Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and duality statements are given. New numerical methods for efficiently solving set optimization problems are provided. Moreover, applications in economics, finance and risk theory are discussed. Summary The objective of this book is to present advances in different areas of variational analysis and set optimization, especially uncertain optimization, optimal control and bilevel optimization. Uncertain optimization problems will be approached from both a stochastic as well as a robust point of view. This leads to different interpretations of the solutions, which widens the choices for a decision-maker given his preferences. Recent developments regarding linear and nonlinear scalarization techniques with solid and nonsolid ordering cones for solving set optimization problems are discussed in this book. These results are useful for deriving optimality conditions for set and vector optimization problems. Consequently, necessary and sufficient optimality conditions are presented within this book, both in terms of scalarization as well as generalized derivatives. Moreover, an overview of existing duality statements and new duality assertions is given. The book also addresses the field of variable domination structures in vector and set optimization. Including variable ordering cones is especially important in applications such as medical image registration with uncertainties. This book covers a wide range of applications of set optimization. These range from finance, investment, insurance, control theory, economics to risk theory. As uncertain multi-objective optimization, especially robust approaches, lead to set optimization, one main focus of this book is uncertain optimization. Important recent developments concerning numerical methods for solving set optimization problems sufficiently fast are main features of this book. These are illustrated by various examples as well as easy-to-follow-steps in order to facilitate the decision process for users. Simple techniques aimed at practitioners working in the fields of mathematical programming, finance and portfolio selection are presented. These will help in the decision-making process, as well as give an overview of nondominated solutions to choose from.
Categories: Business & Economics

Encyclopedia of Optimization

Encyclopedia of Optimization

Pallaschke D, Rolewicz S (1997) Foundations of Mathematical Optimization (Convex Analysis without Linearity). Kluwer, Dordrecht 19. Pinter JD (1996) Global Optimization in Action. Continuous and Lipschitz Optimization: Algorithms, ...

Author: Christodoulos A. Floudas

Publisher: Springer Science & Business Media

ISBN: 9780387747583

Category: Mathematics

Page: 4646

View: 959

The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".
Categories: Mathematics

Generalized Convexity Generalized Monotonicity and Applications

Generalized Convexity  Generalized Monotonicity and Applications

R. Horst and H. Tuy, Global Optimization (Deterministic Approaches), third edition, Springer-Verlag, 1996. H. Konno and T. Kuno, ... I. Singer, Abstract convex analysis, Wiley-Interscience Publication, New York, 1997.

Author: Andrew Eberhard

Publisher: Springer Science & Business Media

ISBN: 9780387236391

Category: Business & Economics

Page: 350

View: 856

In recent years there is a growing interest in generalized convex fu- tions and generalized monotone mappings among the researchers of - plied mathematics and other sciences. This is due to the fact that mathematical models with these functions are more suitable to describe problems of the real world than models using conventional convex and monotone functions. Generalized convexity and monotonicity are now considered as an independent branch of applied mathematics with a wide range of applications in mechanics, economics, engineering, finance and many others. The present volume contains 20 full length papers which reflect c- rent theoretical studies of generalized convexity and monotonicity, and numerous applications in optimization, variational inequalities, equil- rium problems etc. All these papers were refereed and carefully selected from invited talks and contributed talks that were presented at the 7th International Symposium on Generalized Convexity/Monotonicity held in Hanoi, Vietnam, August 27-31, 2002. This series of Symposia is or- nized by the Working Group on Generalized Convexity (WGGC) every 3 years and aims to promote and disseminate research on the field. The WGGC (http://www.genconv.org) consists of more than 300 researchers coming from 36 countries.
Categories: Business & Economics