Shuvomoy Das Gupta

These are my handwritten notes that I took during my graduate studies on the topic of monotone operator theory and operator splitting based on Convex Analysis and Monotone Operator Theory in Hilbert Spaces by Bauschke and Combettes. I hosted the notes on my website but did not provide any external link on the menu for some time as I assumed that no one other than me would read these notes. But during INFORMS annual meeting 2024, Anh Duc Nguyen, who is a brilliant undergraduate student at the National University of Singapore approached me and told me that he read those notes and found those very useful! Hence, I am putting an external link to those notes here, in case someone find them useful in the future.

Chapter 1. Background
Chapter 2. Hilbert Spaces
Chapter 3. Convex Sets
Chapter 4. Convexity and nonexpansiveness
Chapter 5. Fejer Monotonicity and Fixed Point
Chapter 6. Convex Cones and Generalized Interiors
Chapter 7. Support Functions and Polar Sets
Chapter 8. Convex Functions
Chapter 9. Lower Semicontinuous Convex Functions
Chapter 10. Convex Functions Variants
Chapter 11. Convex Variational Problems
Chapter 12. Infimal Convolution
Chapter 13. Conjugation
Chapter 14. Further Conjugation Results
Chapter 15. Fenchel-Rockafellar Duality
Chapter 16. Subdifferentiality
Chapter 17. Gateaux and Frechet Differentiability
Chapter 18. Further Differentiability Results
Chapter 19. Duality in Convex Optimization
Chapter 20. Monotone Operators
Chapter 21. Finer Properties of Monotone Operator
Chapter 23. Resolvents of Monotone Operators
Chapter 24. Sums of Monotone Operators
Chapter 25. Zeros of Sums of Monotone Operators
Chapter 26. Fermat's Rule in Convex Optimization
Chapter 27. Proximal Minimization
Chapter 28. Projection Operators
Chapter 29. Best Approximation Algorithms

These notes are based on the course EE364b

0. Notation
1. Subgradient
2. Subgradient calculus
3. Subgradient methods
4. Subgradient methods for constrained optimization problems
5. Primal-dual subgradient method
6. Introduction to proximal mapping
7. Proximal algorithm
8. Proximal gradient algorithm
9. Projections onto different sets
10. Computing proximal maps and projections
11. Parallel and distributed algorithms
12. Coordinate descent algorithm
13. Revisiting decomposition algorithms
14. Distributed optimization algorithm by circuits
15. Introduction to monotone operators
16. Nonexpansive and contraction mapping
17. Resolvent and Cayley operator
18. Fixed point iteration
19. Proximal point algorithm and method of multipliers
20. Davis-Yin Operator Splitting
21. Operator splitting algorithms, Douglas-Rachford splitting
22. Consensus optimization
23. ADMM from operator splitting
24. ADMM classical derivation