# Lecture Plan

The topic videos can be found on Blackboard under "Learning materials".

Date Topics Reading Slides and code
Introduction, notion and existence of solutions
Week 2 Overview
Formulation of a minimization problem
Definitions of minima
Lower semi-continuity and coercivity
Existence theorems for minima
First order necessary conditions
Second order necessary and sufficient conditions
N&W, Chapters 1, 2.1

Minimisers of Optimisation Problems
General Information Introduction Minima Necessary and Sufficient Optimality Conditions
Week 3 Convex functions
First and second order characterisations
Basic Properties of Convex Functions Convexity
Exact line search methods
Armijo condition
Wolfe conditions
N&W, Chapter 3.1 Descent Methods Gradient Descent Methods Line Search Methods Inexact Line Search Methods
Week 4 Line search methods
Zoutendijk's result
Method for Wolfe conditions
Backtracking line search
Convergence of Steepest Descent
Newton's method
Damped Newton's method
N&W, Chapter 3.1-3.2

Convergence of descent methods with backtracking

N&W, Chapter 3.3
Convergence of Line Search Methods - Zoutendijk's Result Rate of Convergence Convergence Rate of Steepest Descent Newton's Method Question Session Sufficient Decrease and Backtracking
Week 5 Linear Conjugate Gradient Methods
Fletcher–Reeves method
Polak–Ribière method
N&W, Chapter 5.1-2 Conjugate Gradient Methods 1 Conjugate Gradient Methods 2 Conjugate Gradient Methods 3
Week 6 Least Squares problems
Gauss–Newton Method
N&W, Chapter 10.1,10.3 (to page 257) Least-Squares Problems (updated version with a typo corrected)
Theory and basic methods for constrained optimization
Constrained optimisation over convex sets
Feasible directions
First order optimality conditions for convex sets
Projections on convex sets
Normal cones
Optimisation with convex constraints

N&W, Chapter 12.1–12.4.1)
Optimization with Convex Constraints I Optimization with Convex Constraints I - with Notes Optimization with Convex Constraints II
Week 7 Conditions for linear constraints
Tangent cones for nonconvex sets
Necessary conditions for non-convex sets
Linearised feasible directions
Constraint qualifications
LICQ
Farkas' Lemma
KKT-conditions
N&W, Chapter 12.2–12.4 Conditions for linear constraints Concave inequality constraints Tangent Cone and Constraint Qualifications Question Session Question Session with Notes
Week 8 Review Sessions Review Lecture (Covering Lectures 1-5) Review Session by Markus Köbis
Week 9 Second order necessary and sufficient conditions N&W, Chapter 12.5 Second-Order Optimality Conditions for Constrained Optimization
Week 10 Quadratic penalty method
Nonsmooth penalty methods
N&W Chapters 17.1–17.2 Quadratic Penalty Method Nonsmooth penalty methods
Week 11 Augmented Lagrangian method
Penalty methods for inequality constraints
Logarithmic Barrier Methods
N&W, Chapter 17.3
N&W 17.4
N&W 19.6
Augmented Lagrangian & Logarithmic Barrier Method Practical Augmented Lagrangian
Week 12 Quasi-Newton Methods
SR1-method
DFP Method
BFGS Method
N&W, Chapter 6.1,6.2,6.4 Quasi-Newton Methods
Week 13 Easter Break
Week 14 Trust region methods
Exact solution of the trust region problem
N&W Chapter 4.1–4.3 Trust Region Methods Part 1 (updated version with a typo corrected)
Week 15 Cauchy point
Dogleg method
Convergence of trust region methods
N&W Chapter 4.1–4.3 Trust Region Methods Part 2
Advanced methods for constrained optimization
Linear programs
Lagrangian duality
Primal and dual linear programs
N&W Chapter 13.1
N&W Chapter 12.9
Linear Programs Part 1 Linear Programming Example Linear Programs Part 2 (updated version with corrections)
Week 16 Primal-dual interior point method for linear programs
Central path
Path following methods
Direct solution of equality constrained problems
N&W Chapter 14.1

N&W, Chapters 16.1, 16.2
Primal-dual Method Example for Quadratic Programming - Portfolio Optimization Quadratic Programs with Equality Constraints
Week 17 Active set methods
SQP method (equality and inequality constraints)
SQP method (equality constraints)
Newton's method for KKT equations.
Merit functions
Maratos effect
N&W, Chapters 16.5

N&W, Chapters 18.1, 18.3

N&W, Chapters 15.4, 15.5
Quadratic Programs (Direct Methods, cont. and Active Set Methods); SQP Method
1)
12.2 is particularly relevant here.