# Optimisation

## Unit aims

The aim of this unit is to make students acquainted with the main concepts, ideas, methods, tools and techniques of the mathematical optimisation.

## Unit description

Optimisation can be described as the processes of selecting a best solution (or a decision) out of available alternatives. As such, optimisation is involved in a number of human activities and almost all branches of natural sciences.

For example, investors seek to create portfolios avoiding excessive risk and achieving high return rates. Manufactures aim to maximize the efficiency of their production processes. Engineers adjust parameters to optimise the performance of their designs. Physical systems tend to a state of a minimum energy. Molecules in an isolated system tend to react with each other until the total potential energy is minimized. Rays of light follow paths minimising their travel time.

Mathematically speaking, optimisation is the process of minimising (or maximizing) a multivariable function subject to constraints on its variables.

## Relation to other units

This is a new unit for 2018/19 which replaces Optimisation 2 (MATH20600).  You may not take MATH30017 if you have previously taken the Level 5 version of Optimisation (MATH20600).

## Learning objectives

At the end of the unit, the students should:

• understand the basic theoretical aspects of optimisation problems.
• understand the numerical methods for optimisation problems and their properties.
• be able to solve simple optimisation problems by hand.
• be able to solve (relatively) simple optimisation problems numerically.

## Transferable skills

The unit should provide students with good understanding of the theoretical and numeral aspects of mathematical optimisation.

## Syllabus

This unit would be focused on the main theoretical aspects of optimisation problems and on the methods for solving such problems. Regarding the theoretical aspects of optimisation problems, the following will be included:

• (i) Geometric properties of linear programming problems and their solutions.
• (ii) Duality theory for linear programming.
• (iii) Necessary and sufficient conditions for non-linear unconstrained optimisation.
• (iv) Necessary and sufficient conditions for non-linear constrained optimisation.

Regarding the methods for solving optimisation problems, the following will be covered:

• (a) Numerical methods for unconstrained optimisation including gradient algorithm and Newton method.
• (b) Numerical methods for unconstrained optimisation including penalty and barrier algorithms, methods based on the Lagrangian approach and interior-point methods.

## Reading and References

Recommended reading:

• M.S. Bazaraa, H.F. Sherali and C.M. Shetty, Nonlinear Programming – Theory and Algorithms, Wiley 2006
• M.S. Bazaraa, J.J. Jarvis and H.F. Sherali, Linear Programming and Network Flows, Wiley 2009
• D. Bertsekas, Nonlinear Programming, Athena Scientific, 2016
• D. Bertsimas and J.P. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific 1997
• J.F. Bonnans, J.C. Gilbert, C. Lamerechal and C.A. Sagastizabal, Numerical Optimization – Theory and Practical Aspects, Springer 2006
• J. Nocedal and S.P. Wright, Numerical Optimization, Springer 2006

Unit code: MATH30017
Level of study: H/6
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Vladislav Tadic

## Pre-requisites

MATH10003 Analysis 1A, MATH10006 Analysis 1B, MATH11007 Calculus 1, and MATH11005 Linear Algebra and Geometry. MATH20901 Multivariable Calculus is desirable.

None

## Methods of teaching

Lectures, problem sheets and office hours.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

• 80% Exam
• 20% Two computationally-orientated homeworks

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.