Week 1: Introduction and background material - 1
Review of Linear Algebra
Week 2: Background material - 2
Review of Analysis, Calculus
Week 3: Unconstrained optimization
Taylor's theorem, 1st and 2nd order conditions on a stationary point, Properties of descent directions
Week 4: Line search theory and analysis
Wolfe conditions, backtracking algorithm, convergence and rate
Week 5: Conjugate gradient method - 1
Introduction via the conjugate directions method, geometric interpretations
Week 6: Conjugate gradient method - 2
Formulating the conjugate gradient method, expanding subspace theorem, preconditioned conjugate gradient method
Week 7: Nonlinear optimization methods
Nonlinear conjugate gradient method, Convergence and rate for Newton methods, Hessian modification
Week 8: Linear and nonlinear least squares problems
Formulations and techniques for solving least square problems
Week 9: Constrained optimization - Introduction
First order formulation for constrained optimization, equality and inequality constraints, constraint qualification
Week 10: Constrained optimization - KKT conditions
First order necessary conditions (KKT) and a proof sketch of KKT
Week 11: Constrained optimization - Projected gradient descent
Subgradients and projection operators, examples of projected gradient descent
Week 12: Duality in optimization
Geometric interpretations of duality, and sample problem solving using the Lagrangian dual function formulation.
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