Convex optimization
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Convex optimization · 18 tutoring hrs
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Topics typically covered
Standard syllabus
Convex analysis foundations
- Convex sets and convex functions; epigraphs and sublevel sets
- Separation theorems and supporting hyperplanes
- Subgradients and optimality conditions
- Conjugate functions and Fenchel duality
- Strong and strict convexity; smoothness and Lipschitz continuity
Convex optimization problems
- Linear, quadratic, and second-order cone programs
- Semidefinite programming (introduction)
- Duality theory: Slater conditions and KKT for convex problems
- Sensitivity and perturbation analysis
- Generalized inequalities and conic formulations
Algorithms
- Gradient descent and accelerated methods (Nesterov)
- Proximal methods and operator splitting
- Interior-point methods for LP and SDP (overview)
- ADMM and Douglas–Rachford splitting
- Complexity and convergence rates (introduction)
STEM / applied
Applications in science and engineering
- Sparse recovery and compressed sensing (L1 methods)
- Portfolio optimization and risk constraints
- Control: LQR and model predictive control (convex formulations)
- Signal processing: total variation and denoising
- Machine learning: logistic regression, SVMs, and kernel methods (convex views)
Implementation and case studies
- Modeling languages: CVX, CVXPY, or similar
- Scaling to large datasets with stochastic and distributed methods
- Robust optimization and uncertainty sets
- Structure exploitation: sparsity, low rank, and graph patterns
- Debugging infeasibility and unboundedness in conic solvers
Notes
Topics reflect common graduate convex optimization syllabi at US universities, often cross-listed with operations research, EE, or statistics departments.