Advances in Applied Mathematics and Global Optimization (eBook)

Gao has written five books with Springer: Complementarity, Duality and Symmetry in Nonlinear Mechanics.- Advances in Mechanics and Mathematics Volume I.- Advances in Mechanics and Mathematics Volume II.- Nonsmooth/Nonconvex Mechanics.-Duality Principles in Nonconvex Systems. Sherali has written Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems with Springer as well as two other books with John Wiley & Sons.  

Contents 9
Series Preface 11
Preface 13
Constrained Optimism 15
Biographical Summary of Gilbert Strang 17
Education 17
Positions Held 17
Awards and Duties 17
Journal Editor 18
List of Publications of Gilbert Strang 19
Books 19
Papers in Journals and Books 19
Chapter 1 Maximum Flows and Minimum Cuts in the Plane 28
1.1 Introduction 28
1.2 New Questions and Applications 30
1.3 Duality, Coarea, and Cheeger Constants 32
1.4 The Challenge Problems 34
References 37
Chapter 2 Variational Principles and Residual Bounds for Nonpotential Equations 39
2.1 Introduction 39
2.2 Saddle Point Characterizations of Nonsymmetric Linear Equations 40
2.3 Variational Principles for Nonsymmetric Linear Equations 42
2.4 Variational Principles for Semilinear Equations I 44
2.5 Variational Principles for Semilinear Equations II 46
References 48
Chapter 3 Adaptive Finite Element Solution of Variational Inequalities with Application in Contact Problems 50
3.1 Introduction 50
3.2 Model Elliptic Variational Inequality and Its Finite Element Approximation 55
3.3 Dual Formulation and A Posteriori Error Estimation 61
3.4 Residual-Based Error Estimates for the Model Elliptic Variational Inequality 66
3.5 Recovery-Based Error Estimates for the Model Elliptic Variational Inequality 72
3.6 Numerical Example on the Model Elliptic Variational Inequality 79
3.7 Application to a Frictional Contact Problem 89
3.8 Quasistatic Variational Inequalities and Their Discretizations 106
3.9 A Posteriori Error Estimates for the Quasistatic Contact Problem 112
3.10 Numerical Example on the Quasistatic Contact Problem 120
3.11 Concluding Remarks 127
References 127
Chapter 4 Time—Frequency Analysis of Brain Neurodynamics 132
4.1 Introduction 132
4.2 Neurodynamics of the Brain 138
4.3 Design of Experiments 148
4.4 Results 152
4.5 Conclusions and Discussion of Future Research 156
References 160
Chapter 5 Nonconvex Optimization for Communication Networks 162
5.1 Introduction 163
5.2 Internet Congestion Control 165
5.3 Wireless Network Power Control 179
5.4 DSL Spectrum Management 198
5.5 Internet Routing 208
References 218
Chapter 6 Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms 222
6.1 Introduction 222
6.2 Complexity Issues 224
6.3 Optimality Conditions 226
6.4 Algorithms 228
6.5 Multivariate Partition Approach 232
6.6 Applications 238
6.7 Some Interesting Open Problems 239
6.8 Concluding Remarks 241
References 241
Chapter 7 Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes 247
7.1 Introduction 247
7.2 Sufficient Conditions for Bending the Central Path and the Total Curvature 249
7.3 Finding h . 262
7.4 Uniform Distances to the Redundant Hyperplanes 267
7.5 Geometrically Decaying Distances to the Redundant Hyperplanes 272
7.6 Conclusions 279
References 280
Chapter 8 Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization 281
8.1 Introduction 282
8.2 Quadratic Minimization Problems 285
8.3 Canonical Lagrangian Duality Theory 291
8.4 Complementary Variational Principles in Continuum Mechanics 297
8.5 Nonconvex Problems with Double-Well Energy 301
8.6 Canonical Duality Theory in General Nonconvex Systems 309
8.7 Applications to Semilinear Nonconvex Systems 319
8.8 General Constrained Global Optimization Problems 331
8.9 Sequential Canonical Dual Transformation and Solutions to Polynomial Minimization Problems 339
8.10 Concluding Remarks 344
References 344
Chapter 9 Quantum Computation and Quantum Operations 351
9.1 Introduction and Basic Definitions 351
9.2 Completely Positive Maps 354
9.3 Noisy Quantum Channels 359
9.4 Iterations 361
9.5 Fixed Points 363
9.6 Idempotents 365
9.7 Sequential Measurements 366
References 371
Chapter 10 Ekeland Duality as a Paradigm 372
10.1 Introduction 372
10.2 Preliminaries: The Ekeland—Legendre Transform 374
10.3 The Ekeland Duality Scheme 377
10.4 Tools from Nonsmooth Analysis 380
10.5 Ekeland and Legendre Functions 382
10.6 The Fenchel—Rockafellar Duality 387
10.7 The Toland Duality 390
10.8 The Wolfe Duality 392
10.9 The Clarke Duality 395
References 398
Chapter 11 Global Optimization in Practice: State of the Art and Perspectives 400
11.1 Introduction 400
11.2 Global Optimization Strategies 404
11.3 Nonlinear Optimization in Modeling Environments 408
11.4 The LGO Solver Suite and Its Implementations 410
11.5 Illustrative Examples 413
11.6 Global Optimization: Applications and Perspectives 420
11.7 Conclusions 422
References 423
Chapter 12 Two-Stage Stochastic Mixed-Integer Programs: Algorithms and Insights 428
12.1 Introduction 428
12.2 Stage, Scenario, and Test-Set Decomposition 430
12.3 Disjunctive Programming and RLT Cutting Plane Methods for Mixed 0—1 Stochastic Programs 435
12.4 Structural Enumeration Using a Fixed Technology Matrix 451
12.5 Conclusion 455
References 456
Chapter 13 Dualistic Riemannian Manifold Structure Induced from Convex Functions 459
13.1 Introduction 459
13.2 Convex Functions and Riemannian Geometry 460
13.3 The a-Hessian Structure Associated with Convex-Induced Divergence 477
13.4 Summary and Open Problems 483
References 485
Chapter 14 NMR Quantum Computing 487
14.1 Nuclear Magnetic Resonance 487
14.2 Basic Technology Used in Quantum Computation with NMR 495
14.3 Solid-State NMR 514
14.4 Shor’s Algorithm and Its Experimental Realization 523
14.5 Quantum Algorithm for Lattice-Gas Systems 528
14.6 Conclusion 532
Appendix The Homeomorphism from SU(2) to SO(3) 533
References 537