Lasserre J.-B. Moments, positive polynomials and their applications

London: Imperial College Press, 2010. — 371 p.This book is an attempt to convince the reader that one may now consider solving (or at least approximating as closely as desired) some difficult problems that were thought to be out of reach a few years ago, or for which only heuristics were available. Of course, since such problems remain hard, this methodology has some practical limitations mainly due to the size of the original problem, especially in view of the present status of semidefinite programming solvers (used as a black box subroutine to solve each semidefinite relaxation). However, we also indicate how sparsity or symmetries (when present) can be exploited so as to handle problems of larger size. Hence much remains to be done but we hope that this methodology will open the door to a more systematic and efficient treatment of such problems.
We introduce this methodology in a unified manner. To do so we first provide a general numerical scheme for solving (or approximately solving) the abstract GMP with polynomial data. Convergence and several other properties are proved in a very general context. Next, we illustrate in detail the above general methodology when applied to several instances of the GMP in various and very different applications in Global Optimization, Algebra, Probability and Markov Chains, Optimal Control, Mathematical Finance, Multivariate Integration, etc... possibly with ad hoc adjustments if and when necessary. Depending on the problem on hand, additional insights are also provided.
The book is divided into two main parts, and here is a brief chapterby-chapter description of its content. Part I is devoted to the theoretical basis that supports the proposed methodology to solve or approximate the abstract GMP. Part II is devoted to illustrate (and sometimes complement) the methodology for specific instances of the GMP in various domains.