Nonsmooth mechanics and convex optimization:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL [u.a.]
CRC Press
2011
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 425 S. graph. Darst. |
| ISBN: | 9781420094237 |
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| 100 | 1 | |a Kanno, Yoshihiro |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nonsmooth mechanics and convex optimization |c Yoshihiro Kanno |
| 264 | 1 | |a Boca Raton, FL [u.a.] |b CRC Press |c 2011 | |
| 300 | |a XIX, 425 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Contact mechanics |x Mathematics | |
| 650 | 4 | |a Mechanics, Applied |x Mathematics | |
| 650 | 4 | |a Mechanics, Analytic | |
| 650 | 4 | |a Nonsmooth mathematical analysis | |
| 650 | 4 | |a Nonsmooth optimization | |
| 650 | 4 | |a Convex sets | |
| 650 | 4 | |a Duality theory (Mathematics) | |
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Record in the Search Index
| _version_ | 1804145591926652928 |
|---|---|
| adam_text | Titel: Nonsmooth mechanics and convex optimization
Autor: Kanno, Yoshihiro
Jahr: 2011
Contents
Convex Optimization over Symmetric Cone l
Cones, Complementarity, and Conic Optimization 3
1.1 Proper Cones and Conic Inequalities.............. 3
1.1.1 Convex sets and cones.................. 3
1.1.2 Partial order induced by proper cone.......... 5
1.2 Complementarity over Cones.................. 6
1.2.1 Dual cones and self-duality............... 6
1.2.2 Complementarity problems............... 7
1.2.3 Variational inequalities.................. 8
1.2.4 Complementarity over nonnegative orthant...... 9
1.2.5 Overview of complementarity over cones........ 10
1.3 Positive-Semidefinite Cone ................... 11
1.3.1 Positive-semidefinite matrices.............. 12
1.3.2 Inner product of matrices................ 16
1.3.3 Self-duality of positive-semidefinite cone........ 17
1.3.4 Complementarity over positive-semidefinite cone ... 18
1.4 Second-Order Cone ....................... 19
1.4.1 Fundamentals of second-order cone........... 20
1.4.2 Self-duality of second-order cone............ 20
1.4.3 Complementarity over second-order cone........ 22
1.5 Conic Constraints and Their Relationship........... 26
1.6 Conic Optimization ....................... 29
1.6.1 Linear programming................... 30
1.6.2 Semidefinite programming................ 32
1.6.3 Second-order cone programming............ 33
1.7 Notes ............................... 36
Optimality and Duality 39
2.1 Fundamentals of Convex Analysis ............... 39
2.1.1 Convex sets and convex functions............ 40
2.1.2 Monotone functions and convexity........... 41
2.1.3 Closed convex functions................. 44
2.1.4 Subdifferential...................... 45
2.1.5 Conjugate function.................... 47
2.2 Optimality and Duality ..................... 50
2.2.1 Dual problem....................... 50
xvi Contents
2.2.2 Weak duality....................... 51
2.2.3 Strong duality ...................... 53
2.2.4 Optimality condition................... 54
2.2.5 Fenchel duality...................... 55
2.2.6 Lagrangian duality.................... 58
2.2.7 KKT conditions ..................... 61
2.3 Application to Semidefinite Programming........... 63
2.3.1 Fenchel dual problem of SDP.............. 63
2.3.2 Duality and optimality of SDP............. 66
2.3.3 Lagrangian duality of SDP ............... 69
2.4 Notes ............................... 72
3 Applications in Structural Engineering 73
3.1 Compliance Optimization.................... 73
3.1.1 Definition of compliance................. 74
3.1.2 Compliance minimization................ 76
3.1.3 Worst-case compliance and robust optimization .... 79
3.2 Eigenvalue Optimization .................... 81
3.2.1 Eigenvalue optimization of structures ......... 81
3.2.2 SDP formulation..................... 82
3.2.3 Optimality condition................... 84
3.3 Set-Valued Constitutive Law .................. 86
3.3.1 Constitutive law..................... 86
3.3.2 Linear elasticity and Legendre transformation..... 88
3.3.3 Inversion via Fenchel transformation.......... 89
3.3.4 Unilateral contact law and Fenchel transformation . . 91
3.4 Notes ............................... 94
II Cable Networks: An Example in Nonsmooth
Mechanics 97
4 Principles of Potential Energy for Cable Networks 99
4.1 Constitutive law ......................... 99
4.1.1 No-compression model.................. 100
4.1.2 Inclusion form ...................... 101
4.1.3 Variational form..................... 103
4.1.4 Complementarity form.................. 104
4.2 Potential Energy Principles in Convex Optimization Forms . 108
4.2.1 Principle of potential energy in general form...... 108
4.2.2 Principle for large strain................. 112
4.2.3 Principle for linear strain................ 116
4.2.4 Principle for the Green-Lagrange strain........ 117
4.3 More on Cable Networks: Nonlinear Material Law...... 119
4.3.1 Piecewise-linear law................... 120
4.3.2 Piecewise-quadratic law................. 124
Contents xvii
4.4 Notes ............................... 127
5 Duality in Cable Networks: Principles of Complementary
Energy 129
5.1 Duality in Cable Networks (1): Large Strain ......... 130
5.1.1 Embedding to Fenchel form............... 130
5.1.2 Dual problem....................... 131
5.1.3 Duality and optimality.................. 135
5.1.4 Principle of complementary energy........... 139
5.1.5 Existence and uniqueness of solution.......... 145
5.2 Duality in Cable Networks (2): Linear Strain......... 147
5.2.1 Embedding to Fenchel form............... 148
5.2.2 Dual problem....................... 149
5.2.3 Duality and optimality.................. 150
5.2.4 Principle of complementary energy........... 152
5.3 Duality in Cable Networks (3): Green-Lagrange Strain . . . 153
5.3.1 Embedding to Fenchel form............... 153
5.3.2 Dual problem....................... 155
5.3.3 Duality and optimality.................. 157
5.3.4 Principle of complementary energy........... 161
5.4 Notes ............................... 163
III Numerical Methods 165
6 Algorithms for Conic Optimization 167
6.1 Primal-Dual Interior-Point Method .............. 167
6.1.1 Outline of interior-point methods............ 167
6.1.2 Interior-point method for linear programming..... 168
6.1.3 Interior-point method for semidefinite programming . 173
6.2 Reformulation and Smoothing Method ............ 177
6.2.1 Reformulation method.................. 177
6.2.2 Smoothing method.................... 180
6.2.3 Extensions to conic complementarity problems .... 181
6.3 Notes ............................... 183
7 Numerical Analysis of Cable Networks 185
7.1 Cable Networks with Pin-Joints ................ 185
7.2 Cable Networks with Sliding Joints .............. 195
7.3 Form-Finding of Cable Networks................ 200
7.3.1 Form-finding with specified axial forces ........ 201
7.3.2 Special cases....................... 202
7.4 Notes ............................... 206
IV Problems in Nonsmooth Mechanics 209
xviii Contents
8 Masonry Structures 211
8.1 Introduction ........................... 211
8.1.1 Notation.......................... 213
8.2 Principle of Potential Energy for Masonry Structures .... 214
8.2.1 Principle of potential energy .............. 214
8.2.2 Constitutive law..................... 216
8.2.3 Conic optimization formulation............. 221
8.3 Principle of Complementary Energy for Masonry Structures 225
8.3.1 Embedding to Fenchel form............... 225
8.3.2 Dual problem....................... 228
8.3.3 Duality and optimality.................. 232
8.3.4 Principle of complementary energy........... 235
8.4 Numerical Aspects........................ 237
8.4.1 Spatial discretization................... 237
8.4.2 Examples......................... 243
8.5 Notes ............................... 249
9 Planar Membranes 253
9.1 Introduction ........................... 253
9.2 Analysis in Small Deformation ................. 255
9.2.1 Principle of potential energy in small deformation . . 255
9.2.2 Conic optimization formulation............. 259
9.2.3 Principle of complementary energy in small deformation 261
9.3 Principle of Potential Energy for Membranes ......... 264
9.3.1 Constitutive law..................... 264
9.3.2 Principle of potential energy .............. 273
9.4 Principle of Complementary Energy for Membranes ..... 274
9.4.1 Embedding to Fenchel form............... 275
9.4.2 Dual problem....................... 276
9.4.3 Duality and optimality.................. 280
9.4.4 Principle of complementary energy........... 288
9.5 Numerical Aspects........................ 291
9.5.1 Spatial discretization................... 291
9.5.2 Examples......................... 295
9.6 Notes ............................... 305
10 Fractional Contact Problems 311
10.1 Friction Law ........................... 311
10.1.1 Coulomb s law...................... 312
10.1.2 Second-order cone complementarity formulation .... 314
10.2 Incremental Problem ...................... 317
10.2.1 Friction law in incremental problems.......... 318
10.2.2 Contact kinematics.................... 318
10.2.3 Problem formulation................... 321
10.3 Discussions on Various Complementarity Forms ....... 329
Contents xix
10.3.1 On auxiliary variables.................. 329
10.3.2 Maximum dissipation law and its optimality conditions 330
10.3.3 A formulation using projection operator........ 339
10.3.4 Friction law and normality rule............. 340
10.4 Notes ............................... 348
11 Plasticity 351
11.1 Fundamentals of Plasticity ................... 351
11.2 Perfect Plasticity ........................ 356
11.2.1 Classical formulation of flow rule in perfect plasticity . 356
11.2.2 Second-order cone complementarity formulation .... 358
11.3 Plasticity with Isotropic Hardening .............. 362
11.3.1 Linear isotropic hardening law ............. 363
11.3.2 Second-order cone complementarity formulation . . . . 364
11.3.3 Incremental problem................... 367
11.3.4 SOCP formulation of incremental problem....... 370
11.4 Plasticity with Kinematic Hardening ............. 373
11.4.1 Linear kinematic hardening............... 374
11.4.2 Second-order cone complementarity formulation . . . . 375
11.4.3 SOCP formulation of incremental problem....... 377
11.5 Notes ............................... 379
References 381
Index 417
About the Author 425
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| indexdate | 2024-07-09T23:21:54Z |
| institution | BVB |
| isbn | 9781420094237 |
| language | English |
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| physical | XIX, 425 S. graph. Darst. |
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| spelling | Kanno, Yoshihiro Verfasser aut Nonsmooth mechanics and convex optimization Yoshihiro Kanno Boca Raton, FL [u.a.] CRC Press 2011 XIX, 425 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematik Contact mechanics Mathematics Mechanics, Applied Mathematics Mechanics, Analytic Nonsmooth mathematical analysis Nonsmooth optimization Convex sets Duality theory (Mathematics) Konvexe Optimierung (DE-588)4137027-2 gnd rswk-swf Nichtglatte Mechanik (DE-588)4201235-1 gnd rswk-swf Konvexe Optimierung (DE-588)4137027-2 s Nichtglatte Mechanik (DE-588)4201235-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469459&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kanno, Yoshihiro Nonsmooth mechanics and convex optimization Mathematik Contact mechanics Mathematics Mechanics, Applied Mathematics Mechanics, Analytic Nonsmooth mathematical analysis Nonsmooth optimization Convex sets Duality theory (Mathematics) Konvexe Optimierung (DE-588)4137027-2 gnd Nichtglatte Mechanik (DE-588)4201235-1 gnd |
| subject_GND | (DE-588)4137027-2 (DE-588)4201235-1 |
| title | Nonsmooth mechanics and convex optimization |
| title_auth | Nonsmooth mechanics and convex optimization |
| title_exact_search | Nonsmooth mechanics and convex optimization |
| title_full | Nonsmooth mechanics and convex optimization Yoshihiro Kanno |
| title_fullStr | Nonsmooth mechanics and convex optimization Yoshihiro Kanno |
| title_full_unstemmed | Nonsmooth mechanics and convex optimization Yoshihiro Kanno |
| title_short | Nonsmooth mechanics and convex optimization |
| title_sort | nonsmooth mechanics and convex optimization |
| topic | Mathematik Contact mechanics Mathematics Mechanics, Applied Mathematics Mechanics, Analytic Nonsmooth mathematical analysis Nonsmooth optimization Convex sets Duality theory (Mathematics) Konvexe Optimierung (DE-588)4137027-2 gnd Nichtglatte Mechanik (DE-588)4201235-1 gnd |
| topic_facet | Mathematik Contact mechanics Mathematics Mechanics, Applied Mathematics Mechanics, Analytic Nonsmooth mathematical analysis Nonsmooth optimization Convex sets Duality theory (Mathematics) Konvexe Optimierung Nichtglatte Mechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469459&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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