# Lecture plan

Introduction, modelling, and existence of solutions
Week 2 Examples of minimization problems
Formulation of a minimization problem
Feasible set
Definitions of minima
Lower semi-continuity and coercivity
Existence theorems for minima
N&W, Chapters 1, 2.1
Minimisers of Unconstrained Optimisation Problems
Theory and methods for unconstrained optimization
Week 3 Taylor's theorem in higher dimensions
1st and 2nd order necessary and sufficient conditions
Convex sets and functions (basic properties)
N&W, Chapter 2.1
Basic Properties of Convex Functions
Line search methods
Backtracking (Armijo) line search
The Wolfe conditions
N&W, Chapters 3.1, 3.2
Week 4 Backtracking and the Wolfe conditions
Algorithmic treatment of the Wolfe conditions
Zoutendijk's result
Convergence rate of line search methods
N&W, Chapters 3.1-3.3
Week 5 Convergence rate of Newton's method N&W, Chapter 3.3
Conjugate Gradient methods N&W, Chapters 5.1-5.2
Week 6 Nonlinear Conjugate Gradient methods
Fletcher-Reeves and Polack-Ribière
N&W, Chapter 5.2
Quasi-Newton methods
DFP and BFGS methods
Superlinear Convergence of the BFGS method
N&W, Chapters 6.1, 6.4
Summary of Line Search methods
Week 7 Trust-region methods
Cauchy point and dogleg method
Two-dimensional subspace minimisation
Overview of convergence results
N&W, Chapters 4.1-4.3
Non-linear least-squares problems
Gauss-Newton method
Levenberg-Marquardt method
N&W Chapter 10.3
Theory and methods for constrained optimization
Week 8 Basics of constrained optimization
Non-smooth unconstrained and smooth constrained problems
Tangent cones
Feasible and linearized feasible directions
Constraint qualifications
Necessary conditions
KKT-conditions
N&W Chapters 12.1-12.4
For the Monday lecture, I suggest to look in particular into Section 12.2, especially examples 12.4 and 12.5; also: Theorem 12.3 (p. 325).
For the Thursday lecture, I suggest to look through Lemma 12.2 and also recall the implicit function theorem (Calculus II and/or Theorem A.2 in N&W).
Week 9 KKT conditions
Conditions for linear constraints
Conditions for convex problems
Second order necessary and sufficient conditions
N&W Chapters 12.3, 12.5, 12.6
For the Monday lecture I suggest to look into Chapter 12.3 (formulation of the KKT conditions).
On Thursday we will discuss the 2nd order necessary and sufficient conditions covered in Chapter 12.5.
Week 10 Quadratic penalty method for equality constrained optimisation
Nonsmooth penalty methods
Augmented Lagrangian method
Penalty methods for inequality constraints
N&W Chapters 17.1-17.3
On Monday we will discuss the quadratic penalty method described in Chapter 17.1. Note that this method is also required in the project.
Week 11 There will be no lectures this week. I will be available during the usual lecture hours in the usual lecture rooms for questions concerning the project.
Week 12 Easter holidays.
Week 13 Augmented Lagrangian method
Non-smooth penalty methods (overview)
Penalty methods for constrained optimisation problems (overview)
N&W Chapters 17.1-17.3
I suggest to review the definitions of the quadratic penalty method and the Augmented Lagrangian method for the lecture (i.e., equations 17.2 and 17.36, as well as Framework 17.3).
Week 14 Barrier methods for inequality constrained optimisation
Merit functions
Treatment of inequality constraints
N&W Chapters 19.6, 19.1, 18.1-18.4
On Monday we will discuss very briefly the usage of logarithmic barrier functions for inequality constrained optimisation, which is closely connected to the concept of interior point methods. Afterwards we will start the discussion of SQP. I suggest to read 19.6 for the barrier functions and the first parts of 18.1 dealing with problems with equality constraints.
Week 15 Linear optimisation
Basic properties of linear programmes
KKT-conditions and duality
Overview of the simplex method
Interior point methods for linear optimisation
N&W Chapters 13.1-13.3, 12.9, 14.1
I suggest to look briefly through the first pages of the chapter of linear optimisation, which contain the most important notions in linear optimisation apart from duality. Concerning duality, I suggest to read the first two pages of 12.9 both before the lecture on Monday and again for the lecture on Thursday.
Week 16 Interior point methods for linear programming
Basics of quadratic programming
Interior point methods for quadratic programming
N&W, Chapters 14.1, 16.1, 16.4, 16.6
Week 17 Repetition N&W, Chapters 1-6, 10-18

N&W always refers to the textbook J. Nocedal and S. Wright: Numerical Optimization, 2nd Edition. Springer, 2006.