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# Optimisation III

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## Description

Most problems in life are optimisation problems: what is the best design for a racing kayak, how do you get the best return on your investments, what is the best use of your time in swot vac, what is the shortest route across town for an emergency vehicle, what are the optimal release rates from a dam for environmental flows in a river? Mathematical formulations of such optimisation problems might contain one or many independent variables. There may or may not be constraints on those variables. There is always, though, an objective: minimise or maximise some function of the variable(s), subject to the constraints. This course will examine nonlinear mathematical formulations, and will concentrate on convex optimisation problems. Many modern optimisation methods in areas such as design of communication networks, finance, etc, rely on the classical underpinnings covered in this course.

## Objective

To extend the students' existing knowledge of optimisation, by providing an appreciation of the complexities of, and techniques for solving, nonlinear optimisation problems. At the end of this subject, students should be able to analyse the features of an optimisation problem with a view to choosing a suitable algorithm for solving the problem; apply suitable algorithms to one- or multi-dimensional optimisation problems, and critically analyse the results; be aware of the theory behind the algorithms studied, so as to be able to adapt the methods to related problems where appropriate; write computer code for algorithms as studied in class; present a mathematical argument to peers (via tutorial presentations).

## Content

Topics covered are: One-dimensional (line) searches: direct methods, polynomial approximation, methods for differentiable functions; Theory of convex and nonconvex functions relevant to optimisation; Multivariable unconstrained optimisation, in particular, higher-order Newton's Method, steepest descent methods, conjugate direction and conjugate gradient methods; Constrained optimisation, including Kuhn-Tucker conditions and the Gradient Projection Method; Penalty methods.

YearSemesterLevelUnits
2013133
 Matthew RoughanLecturer for this course