05477nam 2200661Ia 450 991078073060332120230721024412.01-282-76000-997866127600061-84816-446-7(CKB)2490000000001686(EBL)1679371(OCoLC)729020365(SSID)ssj0000423402(PQKBManifestationID)11284462(PQKBTitleCode)TC0000423402(PQKBWorkID)10440965(PQKB)10297839(MiAaPQ)EBC1679371(WSP)00000616 (Au-PeEL)EBL1679371(CaPaEBR)ebr10422653(CaONFJC)MIL276000(EXLCZ)99249000000000168620090515d2009 uy 0engur|n|---|||||txtccrMoments, positive polynomials and their applications[electronic resource] /Jean-Bernard LasserreLondon Imperial College Press20091 online resource (384 p.)Imperial College Press optimization series,2041-1677 ;1Description based upon print version of record.1-84816-445-9 Includes bibliographical references and index.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 squares2.5.2 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 Density3.4.1 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 Applications5. 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 case5.7 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 Probability7.1 Upper Bounds on Measures with Moment Conditions 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 appropriatImperial College Press optimization series ;1.Moments method (Statistics)PolynomialsMoments method (Statistics)Polynomials.330.0151Lasserre Jean-Bernard1953-58641MiAaPQMiAaPQMiAaPQBOOK9910780730603321Moments, positive polynomials and their applications3703581UNINA