Nonsmooth Vector Functions and Continuous Optimization

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Springer Science & Business Media, Oct 23, 2007 - Mathematics - 270 pages
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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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Contents

PseudoJacobian Matrices
1
12 PseudoJacobian Matrices
10
13 Nonsmooth Derivatives
14
14 PseudoDifferentials and PseudoHessians of Scalar Functions
23
15 Recession Matrices and Partial PseudoJacobians
35
16 Constructing Stable PseudoJacobians
40
17 Gâteaux and Frechet PseudoJacobians
49
Calculus Rules for PseudoJacobians
57
34 Convex Interior Mapping Theorems
118
35 Metric Regularity and PseudoLipschitzian Property
128
Nonsmooth Mathematical Programming Problems
143
42 SecondOrder Conditions
155
43 Composite Programming
168
44 Multiobjective Programming
186
Monotone Operators and Nonsmooth Variational Inequalities
207
52 Generalized Convex Functions
222

22 The Mean Value Theorem and Taylors Expansions
66
23 A General Chain Rule
82
24 Chain Rules Using Recession PseudoJacobian Matrices
85
25 Chain Rules for Gateaux and Frechet PseudoJacobians
93
Openness of Continuous Vector Functions
98
32 Open Mapping Theorems
110
33 Inverse and Implicit Function Theorems
115
53 Variational Inequalities
230
54 Complementarity Problems
243
Bibliographical Notes
255
References
259
Notations
265
Index
266
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