Essential Mathematics for Convex Optimization

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.
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