Moments, Positive Polynomials and Their Applications

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  • Format: Hardcover
  • Copyright: 2009-11-30
  • Publisher: Imperial College Pr

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Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP). This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials. In the second part, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal control, mathematical finance, multivariate integration, etc., and examples are provided for each particular application.

Table of Contents

Prefacep. vii
Acknowledgmentsp. xiii
Moments and Positive Polynomialsp. 1
The Generalized Moment Problemp. 3
Formulationsp. 4
Duality Theoryp. 7
Computational Complexityp. 10
Summaryp. 12
Exercisesp. 13
Notes and Sourcesp. 13
Positive Polynomialsp. 15
Sum of Squares Representations and Semi-definite Optimizationp. 16
Nonnegative Versus s.o.s. Polynomialsp. 20
Representation Theorems: Univariate Casep. 22
Representation Theorems: Mutivariate Casep. 24
Polynomials Positive on a Compact Basic Semi-algebraic Setp. 28
Representations via sums of squaresp. 28
A matrix version of Putinar's Positivstellensatzp. 33
An alternative representationp. 34
Polynomials Nonnegative on Real Varietiesp. 37
Representations with Sparsity Propertiesp. 39
Representation of Convex Polynomialsp. 42
Summaryp. 47
Exercisesp. 48
Notes and Sourcesp. 49
Momentsp. 51
The One-dimensional Moment Problemp. 53
The full moment problemp. 54
The truncated moment problemp. 56
The Multi-dimensional Moment Problemp. 57
Moment and localizing matrixp. 58
Positive and flat extensions of moment matricesp. 61
The K-moment Problemp. 62
Moment Conditions for Bounded Densityp. 66
The compact casep. 67
The non compact casep. 68
Summaryp. 70
Exercisesp. 71
Notes and Sourcesp. 71
Algorithms for Moment Problemsp. 73
The Overall Approachp. 73
Semidefinite Relaxationsp. 75
Extraction of Solutionsp. 80
Linear Relaxationsp. 86
Extensionsp. 87
Extensions to countably many moment constraintsp. 87
Extension to several measuresp. 88
Exploiting Sparsityp. 9
Sparse semidefinite relaxationsp. 94
Computational complexityp. 96
Summaryp. 97
Exercisesp. 98
Notes and Sourcesp. 99
Proofsp. 99
Proof of Theorem 4.3p. 99
Proof of Theorem 4.7p. 102
Applicationsp. 107
Global Optimization over Polynomialsp. 109
The Primal and Dual Perspectivesp. 110
Unconstrained Polynomial Optimizationp. 111
Constrained Polynomial Optimization: Semidefinite Relaxationsp. 117
Obtaining global minimizersp. 119
The univariate casep. 121
Numerical experimentsp. 122
Exploiting sparsityp. 123
Linear Programming Relaxationsp. 125
The case of a convex polytopep. 126
Contrasting LP and semidefinite relaxationsp. 126
Global Optimality Conditionsp. 127
Convex Polynomial Programsp. 130
An extension of Jensen's inequalityp. 131
The s.o.s.-convex casep. 132
The strictly convex casep. 133
Discrete Optimizationp. 134
Boolean optimizationp. 135
Back to unconstrained optimizationp. 137
Global Minimization of a Rational Functionp. 138
Exploiting Symmetryp. 141
Summaryp. 143
Exercisesp. 144
Notes and Sourcesp. 144
Systems of Polynomial Equationsp. 147
Introductionp. 147
Finding a Real Solution to Systems of Polynomial Equationsp. 148
Finding All Complex and/or All Real Solutions: A Unified Treatmentp. 152
Basic underlying ideap. 155
The moment-matrix algorithmp. 155
Summaryp. 160
Exercisesp. 161
Notes and Sourcesp. 161
Applications in Probabilityp. 163
Upper Bounds on Measures with Moment Conditionsp. 163
Measuring Basic Semi-algebraic Setsp. 168
Measures with Given Marginalsp. 175
Summaryp. 177
Exercisesp. 177
Notes and Sourcesp. 178
Markov Chains Applicationsp. 181
Bounds on Invariant Measuresp. 183
The compact casep. 183
The non compact casep. 185
Evaluation of Ergodic Criteriap. 187
Summaryp. 189
Exercisesp. 190
Notes and Sourcesp. 191
Application in Mathematical Financep. 193
Option Pricing with Moment Informationp. 193
Option Pricing with a Dynamic Modelp. 196
Notation and definitionsp. 197
The martingale approachp. 198
Semidefinite relaxationsp. 200
Summaryp. 202
Notes and Sourcesp. 203
Application in Controlp. 205
Introductionp. 205
Weak Formulation of Optimal Control Problemsp. 206
Semidefinite Relaxations for the OCPp. 210
Examplesp. 212
Summaryp. 215
Notes and Sourcesp. 216
Convex Envelope and Representation of Convex Setsp. 219
The Convex Envelope of a Rational Functionp. 219
Convex envelope and the generalized moment problemp. 220
Semidefinite relaxationsp. 223
Semidefinite Representation of Convex Setsp. 225
Semidefinite representation of co(K)p. 226
Semidefinite representation of convex basic semi-algebraic setsp. 229
Algebraic Certificates of Convexityp. 234
Summaryp. 239
Exercisesp. 239
Notes and Sourcesp. 240
Multivariate Integrationp. 243
Integration of a Rational Functionp. 243
The multivariable casep. 245
The univariate essep. 246
Integration of Exponentials of Polynomialsp. 250
The moment approachp. 251
Semidcfinite relaxationsp. 253
The univariate casep. 254
Maximum Entropy Estimationp. 256
The entropy approachp. 257
Gradient and Hessian computationp. 259
Summaryp. 260
Exercisesp. 262
Notes and Sourcesp. 262
Min-Max Problems and Nash Equilibriap. 265
Robust Polynomial Optimizationp. 265
Robust Linear Programmingp. 267
Robust Semidefinite Programmingp. 269
Minimizing the Sup of Finitely Many Rational Functionsp. 270
Application to Nash Equilibriap. 273
N-player gamesp. 273
Two-player zero-sum polynomial gamesp. 276
The univariate casep. 280
Exercisesp. 251
Notes and Sourcesp. 282
Rounds on Linear PDEp. 285
Linear Partial Differential Equationsp. 285
Notes and Sourcesp. 288
Final Remarksp. 291
Background from Algebraic Geometryp. 293
Fields and Conesp. 293
Idealsp. 295
Varietiesp. 296
Preorderingp. 298
Algebraic and Semi-algebraic Sets over a Real Closed Fieldp. 300
Notes and Sourcesp. 302
Measures, Weak Convergence and Marginalsp. 305
Weak Convergence of Measuresp. 305
Measures with Given Marginalsp. 308
Notes and Sourcesp. 310
Some Basic Results in Optimizationp. 311
Non Linear Programmingp. 311
Semidefinite Programmingp. 313
Infinite-dimensional Linear Programmingp. 316
Proof of Theorem 1.3p. 318
Notes and Sourcesp. 319
The GloptiPoly Softwarep. 321
Presentationp. 321
Installationp. 321
Getting startedp. 322
Descriptionp. 323
Multivariate polynomials (mpol)p. 324
Measures (meas)p. 325
Moments (mom)p. 325
Support constraints (supcen)p. 327
Moment constraints (momcon)p. 327
Floating point numbers (double)p. 328
Solving Moment Problems (msdp)p. 329
Unconstrained minimizationp. 329
Constrained minimizationp. 331
Several measuresp. 335
Notes and Sourcesp. 337
Glossaryp. 339
Bibliographyp. 341
Indexp. 359
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