This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose…mehr
This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.
Jean Bernard Lasserre is Directeur de Recherche at the LAAS laboratory in Toulouse and a member of the Institute of Mathematics of Toulouse (IMT). In 2009 he received the Lagrange Prize, awarded jointly by the Mathematical Optimization Society (MOS) and the Society for Industrial and Applied Mathematics (SIAM). He is the winner of the 2015 INFORMS Optimization Society Khachiyan Prize, awarded for life-time achievements in the area of optimization.
Inhaltsangabe
Preface List of symbols 1. Introduction and messages of the book Part I. Positive Polynomials and Moment Problems: 2. Positive polynomials and moment problems 3. Another look at nonnegativity 4. The cone of polynomials nonnegative on K Part II. Polynomial and Semi-algebraic Optimization: 5. The primal and dual points of view 6. Semidefinite relaxations for polynomial optimization 7. Global optimality certificates 8. Exploiting sparsity or symmetry 9. LP relaxations for polynomial optimization 10. Minimization of rational functions 11. Semidefinite relaxations for semi-algebraic optimization 12. An eigenvalue problem Part III. Specializations and Extensions: 13. Convexity in polynomial optimization 14. Parametric optimization 15. Convex underestimators of polynomials 16. Inverse polynomial optimization 17. Approximation of sets defined with quantifiers 18. Level sets and a generalization of the Löwner-John's problem Appendix A. Semidefinite programming Appendix B. The GloptiPoly software References Index.
Preface List of symbols 1. Introduction and messages of the book Part I. Positive Polynomials and Moment Problems: 2. Positive polynomials and moment problems 3. Another look at nonnegativity 4. The cone of polynomials nonnegative on K Part II. Polynomial and Semi-algebraic Optimization: 5. The primal and dual points of view 6. Semidefinite relaxations for polynomial optimization 7. Global optimality certificates 8. Exploiting sparsity or symmetry 9. LP relaxations for polynomial optimization 10. Minimization of rational functions 11. Semidefinite relaxations for semi-algebraic optimization 12. An eigenvalue problem Part III. Specializations and Extensions: 13. Convexity in polynomial optimization 14. Parametric optimization 15. Convex underestimators of polynomials 16. Inverse polynomial optimization 17. Approximation of sets defined with quantifiers 18. Level sets and a generalization of the Löwner-John's problem Appendix A. Semidefinite programming Appendix B. The GloptiPoly software References Index.
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