Convex functions and optimization methods on Riemannian manifolds:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1994
|
| Schriftenreihe: | Mathematics and its applications
297 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 348 S. graph. Darst. |
| ISBN: | 0792330021 |
Internformat
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| 100 | 1 | |a Udrişte, Constantin |d 1940- |e Verfasser |0 (DE-588)1029686122 |4 aut | |
| 245 | 1 | 0 | |a Convex functions and optimization methods on Riemannian manifolds |c by Constantin Udrişte |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1994 | |
| 300 | |a XV, 348 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 297 | |
| 650 | 7 | |a Convexe functies |2 gtt | |
| 650 | 7 | |a Fonctions convexes |2 ram | |
| 650 | 7 | |a Manifolds |2 gtt | |
| 650 | 7 | |a Optimisation mathématique |2 ram | |
| 650 | 7 | |a Riemann, variétés de |2 ram | |
| 650 | 7 | |a Riemann-vlakken |2 gtt | |
| 650 | 4 | |a Convex functions | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Riemannian manifolds | |
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| 650 | 0 | 7 | |a Optimierung |0 (DE-588)4043664-0 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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|---|---|
| adam_text | CONTENTS
Preface xiii
Chapter 1. Metric properties of Riemannian manifolds 1
§1. Riemannian metric 1
Examples of Riemannian metrics 2
§2. Riemannian connection 4
§3. Differential operators 8
§4. Definite symmetric tensor fields of order two 13
§5. Geodesies and exponential map 15
§6. Metric structure of a Riemannian manifold 21
§7. Completeness of Riemannian manifolds 23
Gordon completeness criterion 24
Nomizu Ozeki theorem 26
Applications to Hamiltonian systems 26
§8. Minimum points of a real function 28
Chapter 2. First and second variations of the p energy of a curve 34
§1. Preliminaries 35
§2. The p energy and the first variation formula 36
§3. Second variation of the p energy 38
§4. Null space of the Hessian of the p energy 41
§5. Index theorem 46
§6. Distance from a point to a closed set 51
§7. Distance between two closed sets 54
Chapter 3. Convex functions on Riemannian manifolds 56
§1. Convex sets in Riemannian manifolds 57
§2. Convex functions on Riemannian manifolds 60
vii
§3. Basic properties of convex functions 66
§4. Directional derivatives and subgradients 71
§5. Convexity of functions of class C 77
§6. Convexity of functions of class C2 81
Convexity of Rosenbrock banana function 83
Examples on the sphere S 84
Examples on Poincare plane 86
Linear affine functions 88
§7. Convex programs on Riemannian manifolds 90
§8. Duality in convex programming 93
§9. Kuhn Tucker theorem on Riemannian manifolds 95
§10. Quasiconvex functions on Riemannian manifolds 97
Nontrivial examples of quasiconvex functions which are
not convex 100
§11. Distance from a point to a closed totally convex set 101
§12. Distance between two closed totally convex sets 105
Chapter 4. Geometric examples of convex functions 108
§1. Example of Greene and Shiohama 109
§2. Example of Wu 110
§3. Examples of Bishop and O Neill 113
§4. Convexity of Busemann functions 116
§5. Construction of Cheeger and Gromoll 121
§6. Preserving the completeness and the convexity 122
Chapter 5. Flows, convexity and energies 128
§1. Flows and energies on Riemannian manifolds 129
§2. General properties of the gradient flow 135
§3. Gradient flow of a convex function 137
§4. Diffeomorphisms imposed by a convex function 141
§5. Energy and flow of an irrotational vector field 146
§6. Energy and flow of a Killing vector field 152
§7. Energy and flow of a conformal vector field 157
Examples of vector fields with dense orbits 164
§8. Energy and flow of an affine vector field 165
§9. Energy and flow of a projective vector field 168
§10. Energy and flow of a torse forming vector field 170
§11. Runge Kutta approximation of the orbits 173
TPascal program for Runge Kutta approximation
of the orbits 176
Chapter 6. Semidefinite Hessians and applications 186
§1. Strongly convex functions on Riemannian manifolds 187
§2. Convex hypersurfaces in Riemannian manifolds 192
§3. Convex functions on Riemannian submanifolds 199
Gradient and Hessian on submanifolds 199
Case of tangent bundle 204
Obata theorem 206
Special hypersurfaces of constant level 207
§4. Convex functions and harmonic maps 208
Examples and applications 211
§5. G connected domains 214
Preliminaries 214
. fii .ne . .ea domains 215
Examples 219
§6. Linear complementarity problems 220
§7. Conservative dynamical systems with convex potential 223
Chapter 7. Minimization of functions on Riemannian manifolds 226
§1. Special properties of the minus gradient flow 227
Minus gradient flow 227
Runge Kutta approximation of a minus gradient line 232
TC program for gradient lines in 3 dimensional space 234
TC program for gradient lines in Poincare plane 237
§2. Numerical approximation of geodesies 238
Approximate solution of Cauchy problem 239
Case of Poincare plane 240
TC program for Poincare geodesies 241
Case of hypersurfaces described by Cartesian implicit
equations 242
TC program for spherical geodesies 244
Approximate solution of boundary value problem 246
TC program for geodesies by boundary conditions 249
§3. General descent algorithm on Riemannian manifolds 252
Descent directions and criteria of stopping 252
Convergence of {grad f(x )} to zero 256
Convergence of {x ) to a critical point 260
§4. Gradient methods on Riemannian manifolds 262
Method of steepest descent 262
Convergence of {grad f(x )} to zero 264
Convergence of {x to a critical point 265
Variants of the gradient method 269
Examples 272
Other gradient methods 276
§5. Generalized Newton method on Riemannian manifolds 279
Radial approximations 279
First construction of the method 279
Second construction of the method 281
Properties of the method 282
§6. General descent algorithm for a constrained minimum 284
Appendices 287
1. Riemannian convexity of functions f:R — » R 287
§0. Introduction 287
§1. Geodesies of (R , g) 287
§2. Geodesies of (R x R , g +1) 289
§3. Convex functions on (R , g) 291
2. Descent methods on the Poincare plane 297
§0. Introduction 297
§1. Poincare plane 297
§2. Linear affine functions on the Poincare plane 298
§3. Quadratic affine functions on the Poincare plane 299
§4. Convex functions on the Poincare plane 300
Examples of hyperbolic convex functions 301
§5. Descent algorithm on the Poincare plane 301
TC program for descent algorithm on Poincare plane (I) 303
TC program for descent algorithm on Poincare plane (II) 305
3. Descent methods on the sphere 311
§1. Gradient and Hessian on the sphere 311
§2. Descent algorithm on the sphere 312
Critical values of the normal stress 313
Critical values of the shear stress 314
TC program for descent method on the unit sphere 316
4. Completeness and convexity on Finsler manifolds 318
§1. Complete Finsler manifolds 318
§2. Analytical criterion for completeness 323
§3. Warped products of complete Finsler manifolds 326
§4. Convex functions on Finsler manifolds 326
References 329
Bibliography 331
Index 341
|
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| indexdate | 2024-07-09T17:41:05Z |
| institution | BVB |
| isbn | 0792330021 |
| language | English |
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| spelling | Udrişte, Constantin 1940- Verfasser (DE-588)1029686122 aut Convex functions and optimization methods on Riemannian manifolds by Constantin Udrişte Dordrecht [u.a.] Kluwer Acad. Publ. 1994 XV, 348 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 297 Convexe functies gtt Fonctions convexes ram Manifolds gtt Optimisation mathématique ram Riemann, variétés de ram Riemann-vlakken gtt Convex functions Mathematical optimization Riemannian manifolds Konvexe Funktion (DE-588)4139679-0 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 s Konvexe Funktion (DE-588)4139679-0 s DE-604 Optimierung (DE-588)4043664-0 s Mathematics and its applications 297 (DE-604)BV008163334 297 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006484450&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Udrişte, Constantin 1940- Convex functions and optimization methods on Riemannian manifolds Mathematics and its applications Convexe functies gtt Fonctions convexes ram Manifolds gtt Optimisation mathématique ram Riemann, variétés de ram Riemann-vlakken gtt Convex functions Mathematical optimization Riemannian manifolds Konvexe Funktion (DE-588)4139679-0 gnd Optimierung (DE-588)4043664-0 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4139679-0 (DE-588)4043664-0 (DE-588)4128295-4 |
| title | Convex functions and optimization methods on Riemannian manifolds |
| title_auth | Convex functions and optimization methods on Riemannian manifolds |
| title_exact_search | Convex functions and optimization methods on Riemannian manifolds |
| title_full | Convex functions and optimization methods on Riemannian manifolds by Constantin Udrişte |
| title_fullStr | Convex functions and optimization methods on Riemannian manifolds by Constantin Udrişte |
| title_full_unstemmed | Convex functions and optimization methods on Riemannian manifolds by Constantin Udrişte |
| title_short | Convex functions and optimization methods on Riemannian manifolds |
| title_sort | convex functions and optimization methods on riemannian manifolds |
| topic | Convexe functies gtt Fonctions convexes ram Manifolds gtt Optimisation mathématique ram Riemann, variétés de ram Riemann-vlakken gtt Convex functions Mathematical optimization Riemannian manifolds Konvexe Funktion (DE-588)4139679-0 gnd Optimierung (DE-588)4043664-0 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Convexe functies Fonctions convexes Manifolds Optimisation mathématique Riemann, variétés de Riemann-vlakken Convex functions Mathematical optimization Riemannian manifolds Konvexe Funktion Optimierung Riemannscher Raum |
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