London School of Economics and Political Science (LSE)

42 Optimisation theory (half unit)

Prerequisites

If taken as part of a BSc degree, 05a Mathematics 1 and 05B Mathematics 2 and
116 Abstract mathematics.

Students are also strongly encouraged to take 41 Advanced Mathematical Analysis.

Aims and objectives
The unit is designed to:
? enable students to obtain a rigourous mathematical background to
optimisation techniques used in areas such as economics and finance;
? enable students to understand the connections between the several aspects of
continuous optimisation, and about the suitability and limitations of
optimisation methods for different purposes.
Learning outcomes
Having followed this unit, students should
? have knowledge and understanding of important definitions, concepts and
results, and how to apply these in different situations
? have knowledge of basic techniques and methodologies in the topics covered
? have basic understanding of the theoretical aspects of the concepts and
methodologies covered
? be able to understand new situations and definitions, including combinations
with elements from different areas covered in the course, investigate their
properties, and relate them to existing knowledge
? be able to think critically and with sufficient mathematical rigour
? be able to express arguments clearly and precisely.
Syllabus
This unit aims to bring together several parts of the wide area of mathematical
optimisation, as encountered in many applied fields. The unit concentrates on
continuous optimisation, and in this sense extends the theory studied in standard
calculus courses. In contrast to the Mathematics 1and Mathematics 2 half-units, the
emphasis in this Optimisation Theory unit will be on the mathematical ideas and
theory used in continuous optimisation. This unit covers the following topics:
? Introduction and review of relevant parts from real analysis, with emphasis on
higher dimensions.
? Weierstrass' Theorem on continuous functions on compact set.
? Review with added rigour of unconstrained optimisation of differentiable
functions.
?
Lagrange’s Theorem on equality constrained optimisation.

? The Kuhn-Tucker Theorem on inequality constrained optimisation.
? Finite and infinite horizon dynamic programming.
Essential reading
Sundaram, R.K. A First Course in Optimization Theory. (Cambridge University
Press, 1996) [ISBN 0521497701].
Assessment
This unit is assessed by a two hour unseen written examination.
All information in this document is subject to confirmation in the Programme Regulations for
degrees and diplomas in Economics, Management, Finance and the Social Sciences that are
reviewed annually. Notice is also given in the Regulations of any units which are being phased
out and students are advised to check unit availability.