2008/09 Undergraduate Module Catalogue

MATH2640 Introduction to Optimisation

10 creditsClass Size: 200

Module manager: Professor C.A. Jones
Email: pmtcaj@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

Pre-requisite qualifications

MATH1331 and MATH1351 and MATH1932, or equivalent.

This module is not approved as an Elective

Module summary

Optimisation ''the quest for the best'' plays a major role in financial and economic theory, e.g. in maximising a company's profits or minimising its production costs. How to achieve such optimality is the concern of this course, which develops the theory and practice of maximising or minimising a function of many variables, either with or without constraints. This course lays a solid foundation for progression onto more advanced topics, such as dynamic optimisation, which are central to the understanding of realistic economic and financial scenarios.

Objectives

To provide a collection of theoretical and algorithmic techniques for determining optimal extrema of arbitrary functions of several variables, either with or without constraints.

On completion of this module, students should be to:
(a) determine the definiteness of quadratic forms;
(b) determine exactly extrema of functions of several variables, with or without constraints, using Lagrange multipliers;
(c) use the calculus of variations to determine extrema of integrals dependent upon one or more arbitrary functions;
(d) determine approximately extrema of functions of several variables, with or without constraints, using a selection of search algorithms.

Syllabus

Several-variable calculus, (6 lectures):
- Representing and visualising functions of 2 variables
- Partial derivatives, total derivatives and chain rule
- Gradient vectors and directional derivatives
- Implicit differentiation, change of variables, Jacobian
- Several-variable Taylor series
- Hessian matrix, stationary points.

Unconstrained optimisation (4 lectures):
- Quadratic forms and eigenvalues
- Definiteness using principal minor tests
- Stationary points, local extrema, unconstrained optimisation, applications in economics
- Cobb-Douglas production functions.

Constrained optimisation (10 lectures):
- Constrained maximisation with equality constraints
- Jacobian derivative
- first-order conditions
- constraint qualifications
- Lagrange multipliers
- constrained quadratic forms
- bordered Hessian
- constrained maximisation with inequality constraints and mixed constraints
- constrained minimisation
- Kuhn-Tucker theory (with applications in economics), economic interpretation of Lagrange multipliers, second-order conditions, bordered Hessian of Lagrangian.

Teaching methods

Delivery type	Number	Length hours	Student hours
Revision Class	3	1.00	3.00
Lecture	20	1.00	20.00
Tutorial	10	1.00	10.00
Private study hours			67.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			100.00

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	four assessed example sheets	10.00
Total percentage (Assessment Coursework)		10.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr	90.00
Total percentage (Assessment Exams)		90.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 16/07/2010

Disclaimer

Browse Other Catalogues

• Undergraduate module catalogue
• Taught Postgraduate module catalogue
• Undergraduate programme catalogue
• Taught Postgraduate programme catalogue

Errors, omissions, failed links etc should be notified to the Catalogue Team.PROD