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Aims:

The module aims to provide students with an overview of the optimization landscape and a practical understanding of most popular optimization techniques and an ability to apply these methods to problems they encounter in their studies e.g., MSc project/dissertation and later in their professional career.

Intended learning outcomes:

On successful completion of the module, a student will be able to:

1. Practically understand a comprehensive set of optimization techniques and their range of applicability.
2. Implement mathematical methods.
3. Apply these techniques to problems they encounter in their studies e.g. MSc project/ dissertation and later in their professional career.
4. Critically evaluate the results, which the methods produced for a given problem.

Indicative content:

This module teaches a comprehensive range of state-of-the-art numerical optimization techniques. It covers a number of approaches to unconstrained and constrained problems, methods for smooth and non-smooth convex problems as well as basics of non-convex optimisation.

The following are indicative of the topics the module will typically cover:

• Mathematical formulation and types of optimisation problems.
• Unconstrained optimization theory e.g.: local minima, first and second order conditions.
• Unconstrained optimization methods e.g.: line-search, trust region, gradient descent, conjugate gradient, Newton, Quasi-Newton, inexact Newton.
• Least Squares problems.
• Constrained optimization theory e.g.: local and global solutions, first order optimality, second order optimality, constraints qualification, equality and inequality constraints, duality, KKT conditions.
• Constrained optimization methods for equality and inequality constraints e.g.: constraints elimination, feasible and infeasible Newton, primal-dual method, penalty, barrier and augmented Lagrangian methods, interior point methods.
• Non-smooth optimization e.g., subgradient calculus, proximal operator, operator splitting, ADMM, non-smooth penalties e.g., L1 or TV.

Requisites:

To be eligible to select this module as an option or elective, a student must: (1) be registered on a programme and year of study for which it is a formally available; (2) have strong competency in Linear Algebra and Analysis; (3) have fluency in matrix calculus; and (4) have working knowledge of Matlab.

The coursework assessment needs to be completed using Matlab and all the solutions are provided in Matlab.