An Introduction to Polynomial and Semi-Algebraic Optimization

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.

'This monograph may be considered as a comprehensive introduction to solving global optimization problems described by polynomials and even semi-algebraic functions. The book is accompanied by a MATLAB® freeware software that implements the described methodology … The well written and extensive introduction may help the reader to knowingly use the book.' Jerzy Ombach, Zentralblatt MATH

• 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.