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 Positivstellensatz -- 2.5.3 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 case -- 3.4.2 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 Polynomials -- 5.1 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 Optimization -- 5.7.1 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 -- 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 -- 8.1.1 The compact case -- 8.1.2 The non compact case -- 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 -- 9.2.1 Notation and definitions -- 9.2.2 The martingale approach -- 9.2.3 Semide finite relaxations.

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 -- 10.3.1 Examples -- 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 -- 11.1.1 Convex envelope and the generalized moment problem -- 11.1.2 Semide finite relaxations -- 11.2 Semide finite Representation of Convex Sets -- 11.2.1 Semide finite representation of co(K) -- 11.2.2 Semide finite representation of convex basic semi-algebraic 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 -- 12.1.1 The multivariable case -- 12.1.2 The univariate case -- 12.2 Integration of Exponentials of Polynomials -- 12.2.1 The moment approach -- 12.2.2 Semide nite relaxations -- 12.2.3 The univariate case -- 12.3 Maximum Entropy Estimation -- 12.3.1 The entropy approach -- 12.3.2 Gradient and Hessian computation -- 12.4 Summary -- 12.5 Exercises -- 12.6 Notes and Sources -- 13. Min-Max Problems and Nash Equilibria -- 13.1 Robust Polynomial Optimization -- 13.1.1 Robust Linear Programming -- 13.1.2 Robust Semide nite Programming -- 13.2 Minimizing the Sup of Finitely Many Rational Functions -- 13.3 Application to Nash Equilibria -- 13.3.1 N-player games -- 13.3.2 Two-player zero-sum polynomial games -- 13.3.3 The univariate case -- 13.4 Exercises -- 13.5 Notes and Sources -- 14. Bounds on Linear PDE -- 14.1 Linear Partial Differential Equations -- 14.2 Notes and Sources -- Final Remarks -- Appendix A Background from Algebraic Geometry -- A.1 Fields and Cones -- A.2 Ideals -- A.3 Varieties -- A.4 Preordering.

A.5 Algebraic and Semi-algebraic Sets over a Real Closed Field -- A.6 Notes and Sources -- Appendix B Measures, Weak Convergence and Marginals -- B.1 Weak Convergence of Measures -- B.2 Measures with Given Marginals -- B.3 Notes and Sources -- Appendix C Some Basic Results in Optimization -- C.1 Non Linear Programming -- C.2 Semide finite Programming -- C.3 Infinite-dimensional Linear Programming . -- C.4 Proof of Theorem 1.3 -- C.5 Notes and Sources -- Appendix D The GloptiPoly Software -- D.1 Presentation -- D.2 Installation -- D.3 Getting started -- D.4 Description -- D.4.1 Multivariate polynomials (mpol) -- D.4.2 Measures (meas) -- D.4.3 Moments (mom) -- D.4.4 Support constraints (supcon) -- D.4.5 Moment constraints (momcon) -- D.4.6 Floating point numbers (double) -- D.5 Solving Moment Problems (msdp) -- D.5.1 Unconstrained minimization -- D.5.2 Constrained minimization -- D.5.3 Several measures -- D.6 Notes and Sources -- Glossary -- Bibliography -- Index.