Nonsmooth Vector Functions and Continuous Optimization (eBook)

V. Jeyakumar, Dinh The Luc (Autoren)

eBook Download: PDF

2007 | 2008
X, 270 Seiten
Springer US (Verlag)
978-0-387-73717-1 (ISBN)

Infos

Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems, as well as variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus, using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function.

---

A recent significant innovation in mathematical sciences has been the progressive use of nonsmooth calculus, an extension of the differential calculus, as a key tool of modern analysis in many areas of mathematics, operations research, and engineering. Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions.The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus by using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function. Such a set of matrices forms a new generalized Jacobian, called pseudo-Jacobian. A direct extension of the classical derivative that follows simple rules of calculus, the pseudo-Jacobian provides an axiomatic approach to nonsmooth calculus, a flexible tool for handling nonsmooth continuous optimization problems.Illustrated by numerous examples of known generalized derivatives, the work may serve as a valuable reference for graduate students, researchers, and applied mathematicians who wish to use nonsmooth techniques and continuous optimization to model and solve problems in mathematical programming, operations research, and engineering. Readers require only a modest background in undergraduate mathematical analysis to follow the material with minimal effort.

Inhalt

Contents 7
Preface 9
1 Pseudo-Jacobian Matrices 11
1.1 Preliminaries 11
1.2 Pseudo-Jacobian Matrices 20
1.3 Nonsmooth Derivatives 24
1.4 Pseudo-Differentials and Pseudo-Hessians of Scalar Functions 33
1.5 Recession Matrices and Partial Pseudo- Jacobians 45
1.6 Constructing Stable Pseudo-Jacobians 50
1.7 Gateaux and Frechet Pseudo-Jacobians 59
2 Calculus Rules for Pseudo- Jacobians 67
2.1 Elementary Rules 67
2.2 The Mean Value Theorem and Taylor’s Expansions 76
2.3 A General Chain Rule 92
2.4 Chain Rules Using Recession Pseudo- Jacobian Matrices 95
2.5 Chain Rules for Gateaux and Frechet Pseudo- Jacobians 103
3 Openness of Continuous Vector Functions 108
3.1 Equi-Invertibility and Equi-Surjectivity of Matrices 108
3.2 Open Mapping Theorems 119
3.3 Inverse and Implicit Function Theorems 124
3.4 Convex Interior Mapping Theorems 127
3.5 Metric Regularity and Pseudo-Lipschitzian Property 137
4 Nonsmooth Mathematical Programming Problems 152
4.1 First-Order Optimality Conditions 152
4.2 Second-Order Conditions 164
4.3 Composite Programming 177
4.4 Multiobjective Programming 195
5 Monotone Operators and Nonsmooth Variational Inequalities 216
5.1 Generalized Monotone Operators 216
5.2 Generalized Convex Functions 231
5.3 Variational Inequalities 239
5.4 Complementarity Problems 252
Bibliographical Notes 264
References 268
Notations 274
Index 275

Produkt-Details

• Erscheint lt. Verlag: 23.10.2007
• Reihe/Serie: Springer Optimization and Its Applications
• Reihe/Serie: Springer Optimization and Its Applications
• Zusatzinfo: X, 270 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Schlagworte: Calculus • Derivative • differential equation • Mathematical Programming • Minimum • Model • Operations Research • Operator • Optimization • programming • SOIA
• ISBN-10: 0-387-73717-0 / 0387737170
• ISBN-13: 978-0-387-73717-1 / 9780387737171