Convex optimization
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Convex Optimization — scope drawn from Boyd & Vandenberghe Convex Optimization (free PDF) and typical US graduate convex optimization syllabi.
Convex analysis foundations
- Convex sets and convex functions — Convex sets and convex functions; epigraphs and sublevel sets
- Separation theorems — Separation theorems
- Supporting hyperplanes — Supporting hyperplanes
- Subgradients and optimality conditions — Subgradients and optimality conditions
- Conjugate functions and Fenchel duality — Conjugate functions and Fenchel duality
- Strong and strict convexity — Strong and strict convexity; smoothness and Lipschitz continuity
Convex optimization problems
- Linear and quadratic forms — Linear and quadratic forms
- Second-order cone programs — Second-order cone programs
- Semidefinite programming (introduction) — Semidefinite programming (introduction)
- Duality theory — Duality theory
- KKT for convex problems — KKT for convex problems
- Sensitivity and perturbation analysis — Sensitivity and perturbation analysis
- Generalized inequalities — Generalized inequalities
- Conic formulations — Conic formulations
Algorithms
- Gradient descent — Gradient descent
- Accelerated methods (Nesterov) — Accelerated methods (Nesterov)
- Proximal methods and operator splitting — Proximal methods and operator splitting
- Interior-point methods for LP — Interior-point methods for LP
- SDP (overview) — SDP (overview)
- ADMM and Douglas–Rachford splitting — ADMM and Douglas–Rachford splitting
- Complexity — Complexity
- Convergence rates (introduction) — Convergence rates (introduction)
Applications in science and engineering
- Sparse recovery — Sparse recovery
- Compressed sensing (L1 methods) — Compressed sensing (L1 methods)
- Portfolio optimization — Portfolio optimization
- Risk constraints — Risk constraints
- Control: LQR and model predictive control (convex formulations) — Control: LQR and model predictive control (convex formulations)
- Signal processing — Signal processing
- Denoising — Denoising
- Machine learning — Machine learning
- Kernel methods (convex views) — Kernel methods (convex views)
Implementation and case studies
- Modeling languages: CVX, CVXPY, or similar — Modeling languages: CVX, CVXPY, or similar
- Scaling to large datasets with stochastic — Scaling to large datasets with stochastic
- Distributed methods — Distributed methods
- Robust optimization and uncertainty sets — Robust optimization and uncertainty sets
- Structure exploitation — Structure exploitation
- Graph patterns — Graph patterns
- Debugging infeasibility — Debugging infeasibility
- Unboundedness in conic solvers — Unboundedness in conic solvers
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Convex analysis foundations
Convex sets and convex functions; epigraphs and sublevel sets
Coming soon - Convex optimization problems
Linear and quadratic forms
Coming soon - Algorithms
Gradient descent
Coming soon - Applications in science and engineering
Sparse recovery
Coming soon - Implementation and case studies
Modeling languages: CVX, CVXPY, or similar
Coming soon
Notes
Graduate Convex Optimization — scope drawn from Boyd & Vandenberghe Convex Optimization (free PDF) and typical US graduate convex optimization syllabi. Topic outline: `content/topics/graduate/convex_optimization.json`.