Moments, positive polynomials and their applications [[electronic resource] /] / Jean-Bernard Lasserre

• Record Nr.: UNINA9910455572703321
• Autore: Lasserre Jean-Bernard <1953->
• Pubbl/distr/stampa: London, : Imperial College Press, 2009
• ISBN: 1-282-76000-9; 9786612760006; 1-84816-446-7
• Descrizione fisica: 1 online resource (384 p.)
• Collana: Imperial College Press optimization series, , 2041-1677 ; ; 1
• Disciplina: 330.0151
• Soggetti: Moments method (Statistics); Polynomials; Electronic books.
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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; 2.5.1 Representations via sums of squares; 2.5.2 A matrix version of Putinar's Positivstellensatz2.5.3 An alternative representation; 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; 3.1.1 The full moment problem; 3.1.2 The truncated moment problem; 3.2 The Multi-dimensional Moment Problem; 3.2.1 Moment and localizing matrix; 3.2.2 Positive and at extensions of moment matrices; 3.3 The K-moment Problem; 3.4 Moment Conditions for Bounded Density; 3.4.1 The compact case3.4.2 The non compact case; 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; 4.5.1 Extensions to countably many moment constraints; 4.5.2 Extension to several measures; 4.6 Exploiting Sparsity; 4.6.1 Sparse semide nite relaxations; 4.6.2 Computational complexity; 4.7 Summary; 4.8 Exercises; 4.9 Notes and Sources; 4.10 Proofs; 4.10.1 Proof of Theorem 4.3; 4.10.2 Proof of Theorem 4.7; Part II Applications; 5. Global Optimization over Polynomials5.1 The Primal and Dual Perspectives; 5.2 Unconstrained Polynomial Optimization; 5.3 Constrained Polynomial Optimization: Semide nite Relaxations; 5.3.1 Obtaining global minimizers; 5.3.2 The univariate case; 5.3.3 Numerical experiments; 5.3.4 Exploiting sparsity; 5.4 Linear Programming Relaxations; 5.4.1 The case of a convex polytope; 5.4.2 Contrasting LP and semide nite relaxations.; 5.5 Global Optimality Conditions; 5.6 Convex Polynomial Programs; 5.6.1 An extension of Jensen's inequality; 5.6.2 The s.o.s.-convex case; 5.6.3 The strictly convex case; 5.7 Discrete Optimization5.7.1 Boolean optimization; 5.7.2 Back to unconstrained 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; 6.3.1 Basic underlying idea; 6.3.2 The moment-matrix algorithm; 6.4 Summary; 6.5 Exercises; 6.6 Notes and Sources; 7. Applications in Probability; 7.1 Upper Bounds on Measures with Moment Conditions
• Sommario/riassunto: 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 <i>Generalized Moment Problem (GMP)</i>. 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 appropriat