Real algebraic geometry and optimization / Thorsten Theobald.

Includes bibliographical references and index.

Semialgebraic sets -- Convexity and positivity -- Polynomial optimization -- Sums of squares -- Semidefinite programming -- Real varieties -- Moment problems -- Polyhedral geometry -- Computational methods and applications.

This graduate-level textbook presents a comprehensive introduction to the interaction between real algebraic geometry and modern optimization theory. The book develops the algebraic, geometric, and computational foundations needed to study polynomial systems, semialgebraic sets, convexity, positivity, and optimization problems arising in mathematics, engineering, computer science, robotics, and data science. The author systematically explains core concepts such as real varieties, sums of squares, semidefinite programming, moment methods, polynomial optimization, and convex algebraic geometry while emphasizing both theoretical structure and algorithmic applications. Numerous examples and applications demonstrate how algebraic-geometric methods can be used to solve optimization problems in practical and interdisciplinary contexts. Designed for graduate students and researchers, the text serves as an accessible introduction to contemporary developments connecting algebraic geometry, convex analysis, and optimization theory.