An Introduction to Polynomial and Semi-Algebraic Optimization

The first comprehensive introduction to the powerful moment approach for solving global optimization problems.

Jean Bernard Lasserre (Author)

9781107060579, Cambridge University Press

Hardback, published 19 February 2015

354 pages, 15 b/w illus. 2 colour illus.
22.9 x 15.2 x 2.1 cm, 0.61 kg

'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 Reviews

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.

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.

Subject Areas: Optimization [PBU], Probability & statistics [PBT], Topology [PBP], Geometry [PBM], Numerical analysis [PBKS], Number theory [PBH]