Moments, Positive Polynomials And Their Applications.

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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Formato: Electrónico eBook
Idioma:Inglés
Publicado: World Scientific 2009.
Temas:
Moments method (Statistics)
Polynomials.
Méthodes des moments (Statistique)
Polynômes.
Polynomials
Electronic resource.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover13;
  • Contents
  • Preface
  • Acknowledgments
  • Part I Moments and Positive Polynomials
  • 1. The Generalized Moment Problem
  • 1.1 Formulations
  • 1.2 Duality Theory
  • 1.3 Computational Complexity
  • 1.4 Summary
  • 1.5 Exercises
  • 1.6 Notes and Sources
  • 2. Positive Polynomials
  • 2.1 Sum of Squares Representations and Semi-de nite Optimization
  • 2.2 Nonnegative Versus s.o.s. Polynomials
  • 2.3 Representation Theorems: Univariate Case
  • 2.4 Representation Theorems: Mutivariate Case
  • 2.5 Polynomials Positive on a Compact Basic Semi-algebraic Set
  • 2.6 Polynomials Nonnegative on Real Varieties
  • 2.7 Representations with Sparsity Properties
  • 2.8 Representation of Convex Polynomials
  • 2.9 Summary
  • 2.10 Exercises
  • 2.11 Notes and Sources
  • 3. Moments
  • 3.1 The One-dimensional Moment Problem
  • 3.2 The Multi-dimensional Moment Problem
  • 3.3 The K-moment Problem
  • 3.4 Moment Conditions for Bounded Density
  • 3.5 Summary
  • 3.6 Exercises
  • 3.7 Notes and Sources
  • 4. Algorithms for Moment Problems
  • 4.1 The Overall Approach
  • 4.2 Semide nite Relaxations
  • 4.3 Extraction of Solutions
  • 4.4 Linear Relaxations
  • 4.5 Extensions
  • 4.6 Exploiting Sparsity
  • 4.7 Summary
  • 4.8 Exercises
  • 4.9 Notes and Sources
  • 4.10 Proofs
  • Part II Applications
  • 5. Global Optimization over Polynomials
  • 5.1 The Primal and Dual Perspectives
  • 5.2 Unconstrained Polynomial Optimization
  • 5.3 Constrained Polynomial Optimization: Semide nite Relaxations
  • 5.4 Linear Programming Relaxations
  • 5.5 Global Optimality Conditions
  • 5.6 Convex Polynomial Programs
  • 5.7 Discrete Optimization
  • 5.8 Global Minimization of a Rational Function
  • 5.9 Exploiting Symmetry
  • 5.10 Summary
  • 5.11 Exercises
  • 5.12 Notes and Sources
  • 6. Systems of Polynomial Equations
  • 6.1 Introduction
  • 6.2 Finding a Real Solution to Systems of Polynomial Equations
  • 6.3 Finding All Complex and/or All Real Solutions: A Uni ed Treatment
  • 6.4 Summary
  • 6.5 Exercises
  • 6.6 Notes and Sources
  • 7. Applications in Probability
  • 7.1 Upper Bounds on Measures with Moment Conditions
  • 7.2 Measuring Basic Semi-algebraic Sets
  • 7.3 Measures with Given Marginals
  • 7.4 Summary
  • 7.5 Exercises
  • 7.6 Notes and Sources
  • 8. Markov Chains Applications
  • 8.1 Bounds on Invariant Measures
  • 8.2 Evaluation of Ergodic Criteria
  • 8.3 Summary
  • 8.4 Exercises
  • 8.5 Notes and Sources
  • 9. Application in Mathematical Finance
  • 9.1 Option Pricing with Moment Information
  • 9.2 Option Pricing with a Dynamic Model
  • 9.3 Summary
  • 9.4 Notes and Sources
  • 10. Application in Control
  • 10.1 Introduction
  • 10.2 Weak Formulation of Optimal Control Problems
  • 10.3 Semide finite Relaxations for the OCP
  • 10.4 Summary
  • 10.5 Notes and Sources
  • 11. Convex Envelope and Representation of Convex Sets
  • 11.1 The Convex Envelope of a Rational Function
  • 11.2 Semide finite Representation of Convex Sets
  • 11.3 Algebraic Certificates of Convexity
  • 11.4 Summary
  • 11.5 Exercises
  • 11.6 Notes and Sources
  • 12. Multivariate Integration
  • 12.1 Integration of a Rational Function
  • 12.2 Integration of Exponentials of Polynomials
  • 12.3 Maximum Entropy Estimation.

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