With an emphasis on timeless essential mathematical background for optimization, this textbook provides a comprehensive and accessible introduction to convex optimization for students in applied mathematics, computer science, and engineering. Authored by two influential researchers, the book covers both convex analysis basics and modern topics such as conic programming, conic representations of convex sets, and cone-constrained convex problems, providing readers with a solid, up-to-date understanding of the field. By excluding modeling and algorithms, the authors are able to discuss the theoretical aspects in greater depth. Over 170 in-depth exercises provide hands-on experience with the theory, while more than 30 'Facts' and their accompanying proofs enhance approachability. Instructors will appreciate the appendices that cover all necessary background and the instructors-only solutions manual provided online. By the end of the book, readers will be well equipped to engage with state-of-the-art developments in optimization and its applications in decision-making and engineering.

Table of Contents:
Preface; Main notational conventions; Part I. Convex Sets in Rn: From First Acquaintance to Linear Programming Duality: 1. First acquaintance with convex sets; 2. Theorems of caratheodory, radon, and helly; 3. Polyhedral representations and Fourier-Motzkin elimination; 4. General theorem on alternative and linear programming duality; 5. Exercises for Part I; Part II. Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets: 6. Separation theorem and geometry of convex sets; 7. Geometry of polyhedral sets; 8. Exercises for Part II; Part III. Convex Functions: 9. First acquaintance with convex functions; 10. How to detect convexity; 11. Minima and maxima of convex functions; 12. Subgradients; 13. Legendre transform; 14. Functions of eigenvalues of symmetric matrices; 15. Exercises for Part III; Part IV. Convex Programming, Lagrange Duality, Saddle Points: 16. Convex programming problems and convex theorem on alternative; 17. Lagrange function and Lagrange duality; 18. Convex programming in cone-constrained form; 19. Optimality conditions in convex programming; 20. Cone-convex functions: elementary calculus and examples; 21. Mathematical programming optimality conditions; 22. Saddle points; 23. Exercises for Part IV; Appendices.