Essential Mathematics for Convex Optimization

A textbook that introduces both convex analysis and modern topics in optimization in a mathematically rigorous yet accessible way.

Fatma Kılınç-Karzan (Author), Arkadi Nemirovski (Author)

9781009510523, Cambridge University Press

Hardback, published 26 June 2025

450 pages
26.1 x 18.7 x 2.8 cm, 1.04 kg

'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

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.

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.

Subject Areas: Optimization [PBU]