Aims: The module gives an introduction to the theory of optimisation as well as the fundamentals of approximation theory.

Content:

1. Recap: necessary and sufficient conditions for local min/max, convex functions and sets, Jensen’s inequality, level sets
2. Iterative algorithms: gradient descent and line search methods
3. Newton's method
4. Linear programming with applications in economics and data science
5. Constrained optimisation
6. Introduction to Neural Networks
7. Approximation theory: polynomial approximation, rational approximation, trigonometric approximation
8. Discrete Fourier and Cosine Transform with applications in imaging and signal processing
9. Introduction to Wavelets

Objectives:

• Understand critical points of multivariable functions
• Apply various techniques to solve nonlinear optimisation problems and understand their applications, in economics and data science
• Use Lagrange multipliers and the Karush–Kuhn–Tucker conditions to solve constrained nonlinear optimisation problems
• Understand the basic concepts of approximation theory
• Obtain an understanding of different approximation techniques used in the digital sciences

Books:

• Endre Sueli and David F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003
• S. Boyd, Convex Optimization, Cambridge University Press, 2004
• J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981
• N. Trefethen, Approximation Theory and Practice