This textbook offers a mathematically rigorous introduction to convex optimization, blending classical theory with modern topics. Its elementary treatment based on linear algebra, calculus, and real analysis, focus on mathematical foundations, conversational tone, and over 170 exercises make this one of the most accessible textbooks on the topic.
This textbook offers a mathematically rigorous introduction to convex optimization, blending classical theory with modern topics. Its elementary treatment based on linear algebra, calculus, and real analysis, focus on mathematical foundations, conversational tone, and over 170 exercises make this one of the most accessible textbooks on the topic.
Fatma K¿l¿nç-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. Y¿ld¿z), 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.
Inhaltsangabe
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.
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.
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