## MATH2640 Introduction to Optimisation

### 10 creditsClass Size: 265

Module manager: Professor Onno Bokhove
Email: o.bokhove@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH1005 or MATH1010 or (MATH1050 and MATH1060) or (MATH1050 and MATH1331), or equivalent.

This module is approved as a discovery module

### Module summary

Optimisation ''the quest for the best'' plays a major role in financial and economic theory, eg 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) determine extrema of functions of several variables subject to inequality constraints, using both classical and Kuhn-Tucker approaches;
(d) apply the theory to a range of problems arising in Mathematical Economics.

### 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):
- 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
- bordered Hessian
- constrained maximisation with inequality constraints and mixed constraints
- constrained minimisation
- Kuhn-Tucker theory
- Application to mean-variance portfolio theory and the Markowitz model

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 11.00 Private study hours 79.00 Total Contact hours 21.00 Total hours (100hr per 10 credits) 100.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment four assessed example sheets 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 85.00 Total percentage (Assessment Exams) 85.00

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