This book presents a novel methodology for solving the Generalized Moment Problem (GMP) when its data involves polynomials and basic semi-algebraic sets. The methodology integrates semidefinite programming with recent advancements in real algebraic geometry to construct a hierarchy of semidefinite relaxations that converge to the optimal value. This approach leverages standard duality in convex optimization to express the relationship between moments and positive polynomials. The second part of the book details the application of this methodology to diverse fields such as global optimization, probability theory, optimal control, mathematical finance, and multivariate integration, providing specific examples for each.

The book introduces a sophisticated methodology for tackling the Generalized Moment Problem, particularly when dealing with polynomial data and semi-algebraic sets. Its approach, combining semidefinite programming with real algebraic geometry, offers a structured way to approximate solutions through a hierarchy of relaxations. The detailed exploration of applications across various mathematical and financial domains suggests a text aimed at researchers and practitioners seeking advanced techniques. The emphasis on duality between moments and positive polynomials highlights a core theoretical contribution.