Optimization and approximation:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Chichester [u.a.]
Wiley
1979
|
| Series: | A Wiley-Interscience publication
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 220 S. |
| ISBN: | 0471997412 |
Staff View
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Record in the Search Index
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| adam_text | Contents
I. Linear Problems 1
1.1 Introduction 1
1.1.1 The relation between approximation and optimization ... 1
I.I .2 A control approximation problem in the heating of metals . 3
I.I .3 Semi infinite optimization in the control of air pollution . . 5
1.1.4 A preview of convex and general nonlinear problems ... 6
1.2 Examples of linear approximation and optimization problems . . 7
1.2.1 Uniform approximation of functions 7
1.2.1.1 The general case 7
1.2.1.2 Approximation with interpolation side conditions . . . 8
1.2.1.3 The discrete case 8
1.2.2 Uniform approximation for the initial boundary value
problem (IBVP) for the heat equation 9
1.2.3 A linear optimization problem derived from a boundary
value problem for the potential equation 11
1.2.4 Linear boundary value problems and uniform approximation . 13
1.2.4.1 The general case 13
1.2.4.2 Problems of monotonic type and one sided
approximation 14
1.2.5 A linear control—approximation problem 15
1.3 The general linear optimization problem 16
1.3.1 Posing the problem; a weak duality theorem and simple
consequences 16
1.3.2 The semi infinite case 18
1.3.3 Semi infinite problems in function spaces 19
1.3.3.1 Application to a boundary value problem for the
potential equation 21
1.3.3.2 Application to a linear boundary value problem of
monotonic type 22
1.3.4 Counter examples to the validity of general existence and
duality statements 26
1.3.4.1 Unsolvability of a semi infinite problem 26
1.3.4.2 Occurrence of a duality gap 27
1.3.5 Bibliographical remarks 30
ix
X
1.4 Existence and duality theorems 31
1.4.1 Double dualization of an optimization problem 31
1.4.2 Sub consistency and normality of an optimization problem. . 32
1.4.3 General existence and duality theorems 35
1.4.4 Existence theorems for the dual problem 39
1.4.4.1 Dualization of the existence theorem in Section 1.4.3 . . 39
1.4.4.2 Application to the semi infinite problem 40
1.4.4.3 A direct existence theorem for the dual problem ... 41
1.4.5 Bibliographical remarks 43
1.5 Applications 45
1.5.1 Semi infinite problems in function spaces 45
1.5.2 Uniform approximation of functions 48
1.5.3 One sided uniform approximation 55
1.5.4 Application to a boundary value problem for the potential
equation 59
1.5.5 A control—approximation problem in heat conduction ... 61
II. Convex Problems 78
II. 1 Examples of convex approximation and optimization problems . 78
II. 1.1 The general linear approximation problem 78
II. 1.2 Optimal error estimates for linear operator equations ... 80
II.1.2.1 Defect estimates 80
II. 1.2.2 Operator estimates 83
II. 1.3 A problem of optimal control 87
11.2 Convex functions 88
11.2.1 Convex functional 88
11.2.2 Convex mappings 91
11.2.3 Existence statements for linear control problems .... 94
11.3 The general convex optimization problem: existence and
duality theorems 96
11.3.1 Posing the problem, existence theorems, subconsistency . . 96
11.3.2 The dual problem 99
11.3.3 General existence and duality theorems 101
11.3.4 The linear case 104
11.3.5 Bibliographical remarks 105
11.4 Min—sup, max inf and saddle point theorems 106
11.4.1 The optimization problem as a min sup problem .... 106
11.4.2 The dual problem as a max—inf problem 108
11.4.3 The equivalence of the min—sup = max—inf statement
with a saddle point statement 110
11.4.4 The generalized theorem of Kuhn and Tucker Ill
11.4.5 Existence theorems for the dual problem 112
11.4.6 Bibliographical remarks 113
11.5 Application to approximation problems 114
II.5.1 Existence theorems in the case of convex approximation
problems in normed vector spaces 114
i
xi
11.5.2 Uniform approximation of functions 115
H.5.2.1 The general convex case 115
H.5.2.2 The general linear case 118
11.5.3 An approximation problem with a mixed norm 119
11.5.4 Calculation of the minimal deviation for a convex
approximation problem 122
II.6 Convex optimization problems in function spaces 124
11.6.1 Posing the problem and characterizing the optimality . . . 124
11.6.2 A mixed linear convex problem 129
H.6.3 Applications 133
11.6.3.1 Uniform linear approximation with interpolation . . . 133
11.6.3.2 A semi infinite problem in the control of air pollution . 137
III. Nonlinear Problems 144
IH.l Some examples of nonlinear approximation and optimization
problems 144
III.1.1 Nonlinear approximation in normed vector spaces .... 144
IH.l.1.1 General remarks 144
III. 1.1.2 Uniform approximation of functions 144
III.1.1.3 General rational approximation 145
IH.l .2 One and two sided approximation in nonlinear boundary
value problems 146
111.1.2.1 The general case 146
111.1.2.2 An example 149
HI. 1.3 Defect estimates for nonlinear boundary value problems . . 150
III.2 Minimizing convex functionals on arbitrary sets 154
111.2.1 Tangent cones in normed vector spaces 154
111.2.2 Necessary conditions for minimal points of convex
functionals on arbitrary sets 156
IH.2.2.1 A general theorem 156
IH.2.2.2 Application to approximation in normed spaces . . . 157
111.2.2.3 Application to uniform approximation of functions. . 159
IH.2.2.4 Application to Z,! approximation 162
111.2.3 Bibliographical remarks 164
III .3 Nonlinear optimization with an infinite number of side
conditions 165
111.3.1 Necessary conditions for minimal points 165
III.3.1.1 The general case 165
HI.3.1.2 The differentiable case 167
111.3.2 Sufficient conditions for minimal points 171
IH.3.3 Application to nonlinear uniform approximation .... 173
IH.3.4 Semi infinite nonlinear optimization 177
III.3.5 Bibliographical remarks 180
IV. Appendix: Mathematical Aids 184
IV.l Convex cones and linear mappings 184
xii
IV. 1.1 Convex cones and partial orderings 184
IV. 1.2 The (topological) dual space 186
IV.1.3 Linear mappings 189
IV.2 Properties of convex cones and representation of positive
linear forms 190
IV.2.1 Closedness of convex cones 190
IV.2.2 Adjoint cones 192
IV.2.3 Representation of positive linear forms on vector spaces of
continuous functions 194
IV.2.4 Representation of continuous linear forms on vector spaces
of continuous functions 197
IV.3 Convex Sets 200
IV.3.1 Algebraic and topological properties 200
IV.3.2 Separation theorems 202
FV.3.3 Weak convergence 203
References 209
Index 216
|
| any_adam_object | 1 |
| author | Krabs, Werner 1934-2017 |
| author_GND | (DE-588)120318741 |
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| dewey-full | 515 511/.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis 511 - General principles of mathematics |
| dewey-raw | 515 511/.4 |
| dewey-search | 515 511/.4 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-07-09T15:42:56Z |
| institution | BVB |
| isbn | 0471997412 |
| language | English German |
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| physical | XII, 220 S. |
| publishDate | 1979 |
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| series2 | A Wiley-Interscience publication |
| spelling | Krabs, Werner 1934-2017 Verfasser (DE-588)120318741 aut Optimierung und Approximation Optimization and approximation W. Krabs Chichester [u.a.] Wiley 1979 XII, 220 S. txt rdacontent n rdamedia nc rdacarrier A Wiley-Interscience publication Approximation, Théorie de l' Approximation, théorie de l' ram Optimisation mathématique Optimisation mathématique ram Approximation theory Mathematical optimization Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Approximation (DE-588)4002498-2 s Optimierung (DE-588)4043664-0 s DE-604 Approximationstheorie (DE-588)4120913-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484642&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krabs, Werner 1934-2017 Optimization and approximation Approximation, Théorie de l' Approximation, théorie de l' ram Optimisation mathématique Optimisation mathématique ram Approximation theory Mathematical optimization Approximationstheorie (DE-588)4120913-8 gnd Approximation (DE-588)4002498-2 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4120913-8 (DE-588)4002498-2 (DE-588)4043664-0 |
| title | Optimization and approximation |
| title_alt | Optimierung und Approximation |
| title_auth | Optimization and approximation |
| title_exact_search | Optimization and approximation |
| title_full | Optimization and approximation W. Krabs |
| title_fullStr | Optimization and approximation W. Krabs |
| title_full_unstemmed | Optimization and approximation W. Krabs |
| title_short | Optimization and approximation |
| title_sort | optimization and approximation |
| topic | Approximation, Théorie de l' Approximation, théorie de l' ram Optimisation mathématique Optimisation mathématique ram Approximation theory Mathematical optimization Approximationstheorie (DE-588)4120913-8 gnd Approximation (DE-588)4002498-2 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Approximation, Théorie de l' Approximation, théorie de l' Optimisation mathématique Approximation theory Mathematical optimization Approximationstheorie Approximation Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484642&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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