Convex Analysis
This course serves as an introduction to convex sets, convex functions, and convex optimization. The applications of such material are innumerable and are essential knowledge for an understanding of optimization theory and computation, machine learning, optimal transport, fixed-point theory, and many other fields. In this class we try and strike a balance between applied and pure mathematics. We cover essential information needed for applications such as linear programming, duality theorems of convex optimization and minimax theorems. On the theoretical side we will encounter deep theorems about the first- and second-order differentiability of convex functions and some fundamental results on convex analysis in the infinite dimensional case such as the Hahn-Banach theorem, Krein-Milman theorem, and Choquet theory.
Lecturer
Date
16th September ~ 16th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 15:20 - 16:55 | Shuimo | ZOOM 09 | 230 432 7880 | BIMSA |
Prerequisite
Linear algebra, real analysis, measure theory
Syllabus
Week 1: Convex sets, convex functions, Jensen's inequality, boundaries of convex sets, projection theorems
Week 2: Hyperplane separation theorems, supporting hyperplane theorem, Carathedory theorem, extreme points
Week 3: Unbounded convex sets and recession cones, Support functions
Week 4: Continuity properties of convex functions, subgradients, differential continuity,
Week 5: Legendre transform, monotonicity and cyclical monotonicity (Rockafellar's theorem)
Week 6: Polyhedral convexity and linear programming, minimum of convex functions
Week 7: Lagrange multipliers, Fenchel duality
Week 8: Maximum of convex functions, minimax problems, minimax theorems
Week 9: Differentiability of convex functions, Rademacher's theorem
Week 10: Twice differentiability of convex functions, Aleksandrov's theorem
Week 11: Convexity in infinite-dimensional spaces, Hahn-Banach theorem
Week 12: Closest point projection, Krein-Milman theorem, Choquet theorem
Week 13: Convex sets and extreme points in infinite dimensions
Week 2: Hyperplane separation theorems, supporting hyperplane theorem, Carathedory theorem, extreme points
Week 3: Unbounded convex sets and recession cones, Support functions
Week 4: Continuity properties of convex functions, subgradients, differential continuity,
Week 5: Legendre transform, monotonicity and cyclical monotonicity (Rockafellar's theorem)
Week 6: Polyhedral convexity and linear programming, minimum of convex functions
Week 7: Lagrange multipliers, Fenchel duality
Week 8: Maximum of convex functions, minimax problems, minimax theorems
Week 9: Differentiability of convex functions, Rademacher's theorem
Week 10: Twice differentiability of convex functions, Aleksandrov's theorem
Week 11: Convexity in infinite-dimensional spaces, Hahn-Banach theorem
Week 12: Closest point projection, Krein-Milman theorem, Choquet theorem
Week 13: Convex sets and extreme points in infinite dimensions
Reference
Convex Analysis by R. Tyrrell Rockafellar, Convex Optimization Theory by Dimitri Bertsekas, Measure Theory and Fine Properties of Functions by Lawrence Evans and Ronald Gariepy, Functional Analysis by Peter D. Lax
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
My research mostly consists of using tools of analysis and numerical analysis to investigate and compute solutions of problems in optimal transport with “unusual” cost functions. Applications of the mathematical work include optics inverse problems, computational mesh generation, sampling, and optimal control. I completed my Ph.D. thesis on numerical methods for fully nonlinear elliptic PDEs arising in optimal transport in 2022 working under Brittany Hamfeldt at the New Jersey Institute of Technology. From 2022 to 2025 I worked as a postdoc at the University of Texas at Austin under the supervision of Richard Tsai. I joined BIMSA in late May, 2025.