Multivalued Analysis and Nonlinear Programming Problems with Perturbations

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Springer Science & Business Media, Dec 31, 2002 - Mathematics - 205 pages
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The book presents a treatment of topological and differential properties of multivalued mappings and marginal functions. In addition, applications to sensitivity analysis of nonlinear programming problems under perturbations are studied. Properties of marginal functions associated with optimization problems are analyzed under quite general constraints defined by means of multivalued mappings. A unified approach to directional differentiability of functions and multifunctions forms the base of the volume. Nonlinear programming problems involving quasidifferentiable functions are considered as well. A significant part of the results are based on theories and concepts of two former Soviet Union researchers, Demyanov and Rubinov, and have never been published in English before.  It contains all the necessary information from multivalued analysis and does not require special knowledge, but assumes basic knowledge of calculus at an undergraduate level.
 

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Contents

BASIC NOTATION
1
BASIC CONCEPTS AND PROBLEMS OF MULTIVALUED ANALYSIS
5
12 Convex Functions
9
13 Topological and Differential Properties of Convex Functions
12
2 Elements of Nonsmooth Analysis and Optimality Conditions
14
22 Directional Derivatives
15
23 Clarke Subdifferentials
16
3 Quasidifferentiable Functions and Problems
18
2 Weakly Uniformly Differentiable Multivalued Mappings
84
3 Strongly Differentiable Mappings and Directional Differentiability of Marginal Functions
90
32 Directional differentiability of marginal functions
93
SENSITIVITY ANALYSIS
97
1 Stability Properties of Optimal Solutions in Mathematical Programming Problems
98
2 Regular Multivalued Mappings
102
21 Regularity Conditions
103
22 Rregular Mappings
106

32 Necessary Optimality Conditions
22
TOPOLOGICAL AND DIFFERENTIAL PROPERTIES OF MULTIVALUED MAPPINGS
27
12 Marginal Functions
32
13 Pseudolipschitz and Pseudoholder Continuity of Multivalued Mappings
37
14 Properties of Convex Mappings
39
15 Closed convex processes
41
2 Directional Differentiability of Multivalued Mappings
43
22 Description of Derivatives of Multivalued Mappings in Terms of the Distance Function
46
23 Firstorder Approximations of Multivalued Mappings
50
24 Some Properties of Derivatives of Multivalued Mappings
51
3 Lemma About the Removal of Constraints
54
SUBDIFFERENTIALS OF MARGINAL FUNCTIONS
59
12 Pseudolipschitz Continuity and Metrical Regularity
68
2 Locally Convex Mappings
71
22 Subdifferentials of Marginal Functions for Locally Convex Multivalued Mappings
73
DIRECTIONAL DERIVATIVES OF MARGINAL FUNCTIONS
77
23 The Linear Tangent Cone and Derivatives of Regular Multivalued Mappings
114
24 Subdifferentials of Marginal Functions in Regular Problems
117
25 Secondorder Derivatives of Mappings
119
26 Directional Regularity
120
3 FirstOrder Directional Derivatives of Optimal Value Functions and Sensitivity Analysis of Suboptimal Solutions
128
31 General Case
129
32 Directional Derivatives of Optimal Value Functions in Nonlinear Programming Problems
137
33 Holder Behaviour of Optimal Solutions and Directional Differentiability of Optimal Value Functions in Rregular Problems
148
34 Problems with Vertical Perturbations
165
35 Quasidifferentiable Programming Problems
169
4 SecondOrder Analysis of the Optimal Value Function and Differentiability of Optimal Solutions
175
Bibliographical Comments
187
References
191
Index
203
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About the author (2002)

Bernd Luderer: Chair of Business Mathematics, Faculty of Mathematics, Chemnitz University of Technology, Germany Leonid Minchenko: Chair of Informatics, Byelorussian State University of Informatics & Radioelectronics, Byelorussia Tatyana Satsura: Chair of Informatics, Byelorussian State University of Informatics & Radioelectronics, Byelorussia

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