An Introduction to Polynomial and Semi-Algebraic Optimization
Published on 19. February 2015
Book
Hardback
354 pages
978-1-107-06057-9 (ISBN)
Description
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.
Reviews / Votes
'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 (R) 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 'This book provides an accessible introduction to very recent developments in the field of polynomial optimisation, i.e., the task of finding the infimum of a polynomial function on a set defined by polynomial constraints ... Every chapter contains additional exercises and a guide to the (free) Matlab software GloptiPoly. Therefore, this really well-written book provides an ideal introduction for individual learning and is well suited as the basis for a course on polynomical optimisation. Cordian Riener, Mathematical ReviewsMore details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
Professional and scholarly
Illustrations
Worked examples or Exercises; 5 Halftones, unspecified; 2 Halftones, color; 10 Line drawings, unspecified
Dimensions
Height: 235 mm
Width: 157 mm
Thickness: 24 mm
Weight
664 gr
ISBN-13
978-1-107-06057-9 (9781107060579)
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Schweitzer Classification
Other editions
Additional editions
Jean Bernard Lasserre
An Introduction to Polynomial and Semi-Algebraic Optimization
E-Book
08/2016
Cambridge University Press
€41.99
Available for download
Jean Bernard Lasserre
An Introduction to Polynomial and Semi-Algebraic Optimization
Book
02/2015
Cambridge University Press
€97.40
Shipment within 15-20 days
Jean Bernard Lasserre
Introduction to Polynomial and Semi-Algebraic Optimization
E-Book
02/2015
Cambridge University Press
€36.99
Available for download
Person
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.
Content
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 Loewner-John's problem; Appendix A. Semidefinite programming; Appendix B. The GloptiPoly software; References; Index.