Essential Mathematics for Convex Optimization

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.

About the Author :
Fatma Kılınç-Karzan is a Professor of Operations Research at Tepper School of Business, Carnegie Mellon University. She was awarded the 2015 INFORMS Optimization Society Prize for Young Researchers, the 2014 INFORMS JFIG Best Paper Award (with S. Yıldız), and an NSF CAREER Award in 2015. Her research focuses on foundational theory and algorithms for convex optimization and structured nonconvex optimization and their applications in optimization under uncertainty, machine learning, and business analytics. She has been an elected member on the councils of the Mathematical Optimization Society and INFORMS Computing Society, and has served on the editorial boards of MOS-SIAM Book Series on Optimization, MathProgA, MathOR, OPRE, SIAM J Opt, IJOC, and OMS. Arkadi Nemirovski is the John P. Hunter, Jr. Chair and Professor of Industrial and Systems Engineering at Georgia Tech. He has co-authored six optimization textbooks including this one, and he has received many rewards for his contributions to the field, including the 1982 MPS-SIAM Fulkerson Prize (with L. Khachiyan and D. Yudin), the 1991 MPS-SIAM Dantzig Prize (with M. Grotschel), the 2003 INFORMS John von Neumann Theory Prize (with M. Todd), the 2019 SIAM Norbert Wiener Prize (with M. Berger), and the 2023 WLA Prize (with Yu. Nesterov). His research focuses on convex optimization (algorithmic design and complexity analysis, optimization under uncertainty, engineering applications) and nonparametric statistics. He is a member of the National Academy of Engineering, the American Academy of Arts and Sciences, and National Academy of Sciences.

Review :
'This new book by Fatma Kılınç-Karzan and Arkadi Nemirovski is an important contribution to the field of optimization, offering valuable insights for both theoretical research and practical applications. This thorough volume starts with the basics of convex analysis and extends to recent developments in cone-constrained convex problems. The authors include many interesting exercises that help expand on the topics discussed. Additionally, the appendices contain useful supplementary materials that enhance the overall value of the book' Yurii Nesterov, Professor at Corvinus University of Budapest, Emeritus Professor at Catholic University of Louvain, Belgium 'This is a well-structured textbook on the mathematical foundations of convex optimization. It focuses on the structure of convex sets and functions, separation theorems, subgradients, and the theory of duality. The treatment is rigorous but readable, balancing clarity with depth.' Osman Güler, University of Maryland, Baltimore County