Nonsmooth Vector Functions and Continuous Optimization

By: V. Jeyakumar, Dinh The Luc

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Hardcover | 25 October 2007

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280 Pages

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23.5 x 15.88 x 1.91

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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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Optimisation Linear Programming Applied Mathematics Business & Management Operational Research Calculus of Variations

Preface	p. IX
Pseudo-Jacobian Matrices	p. 1
Preliminaries	p. 1
Pseudo-Jacobian Matrices	p. 10
Nonsmooth Derivatives	p. 14
Pseudo-Differentials and Pseudo-Hessians of Scalar Functions	p. 23
Recession Matrices and Partial Pseudo-Jacobians	p. 35
Constructing Stable Pseudo-Jacobians	p. 40
Gateaux and Frechet Pseudo-Jacobians	p. 49
Calculus Rules for Pseudo-Jacobians	p. 57
Elementary Rules	p. 57
The Mean Value Theorem and Taylor's Expansions	p. 66
A General Chain Rule	p. 82
Chain Rules Using Recession Pseudo-Jacobian Matrices	p. 85
Chain Rules for Gateaux and Frechet Pseudo-Jacobians	p. 93
Openness of Continuous Vector Functions	p. 99
Equi-Invertibility and Equi-Surjectivity of Matrices	p. 99
Open Mapping Theorems	p. 110
Inverse and Implicit Function Theorems	p. 115
Convex Interior Mapping Theorems	p. 118
Metric Regularity and Pseudo-Lipschitzian Property	p. 128
Nonsmooth Mathematical Programming Problems	p. 143
First-Order Optimality Conditions	p. 143
Second-Order Conditions	p. 155
Composite Programming	p. 168
Multiobjective Programming	p. 186
Monotone Operators and Nonsmooth Variational Inequalities	p. 207
Generalized Monotone Operators	p. 207
Generalized Convex Functions	p. 222
Variational Inequalities	p. 230
Complementarity Problems	p. 243
Bibliographical Notes	p. 255
References	p. 259
Notations	p. 265
Index	p. 267
Table of Contents provided by Ingram. All Rights Reserved.

Nonsmooth Vector Functions and Continuous Optimization

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