Optimization

What you will study

The module is divided into three blocks of work: solutions of non-linear equations, systems of linear and non-linear equations and mathematical modelling; linear and integer programming; and non-linear optimization for unconstrained and constrained minimization problems. You will be expected to run given computer programs as part of your study, but you will not be required to write any computer programs.

In the broad area of operational research, the module will enable you to formulate a real problem in mathematical terms; to recognise whether the problem can be solved numerically; to choose a suitable method; to understand the conditions required for the method to work; to evaluate the results and to estimate their accuracy and their sensitivity to changes in the data.

Optimization is a practical subject, although it is supported by a growing body of mathematical theory. Problems that require the creation of mathematical models and their numerical solutions arise in science, technology, business and economics as well as in many other fields. Creating and solving a mathematical model usually involves the following main stages:

• formulation of the problem in mathematical terms: this is the creation of a mathematical model
• devising a method of obtaining a numerical solution from the mathematical model
• making observations of the numerical quantities relevant to the solution of the problem
• calculating the solution, usually with a computer or at least with a scientific calculator
• interpreting the solution in relation to the real problem
• evaluating the success or failure of the mathematical model.

Many of the problems discussed in the module arise in operational research and optimization: for example, how to get the most revenue from mining china clay when there is a choice of several mines. In this example the mathematical model consists of a set of linear inequalities defining the output from each mine, the number of mines that can be worked, the correct blend of clay and the total amount of clay mined each year. The method of solving the problem uses mixed linear and integer programming; the numerical data that need to be observed include the financial implications of opening a mine, the number of mines that can be worked with the labour force, and the quality of clay from potential mines. These data will be fed into a computer, which will combine them with the chosen method of solving the equations to produce solutions consisting of outputs from each mine in each year of operation.

This module examines all the stages but concentrates on: the first stage, creating the mathematical model; the second stage, devising a method; the fourth stage, calculating numerical solutions; and the fifth stage, interpreting the solution. Each of the three blocks of work takes about ten weeks of study:

• Block I – Direct and iterative methods of solving single non-linear equations, systems of linear equations and systems of non-linear equations; mathematical modelling; errors in numerical processes, convergence, ill-conditioning and induced instability.
• Block II – Formulation and numerical solution of linear programming problems using the revised simplex method; formulation of integer programming problems and the branch and bound method of solution; sensitivity analysis.
• Block III – Formulation and numerical solution of unconstrained and constrained non-linear optimization problems using, among others, the DFP and BFGS methods with line searches; illustrative applications.

You will learn

Successful study of this module should enhance your skills in:

• mathematical modelling
• operational research
• linear programming and non-linear optimization methods
• the use of iterative methods in problem solving
• the use of Computer Algebra Packages for problem solving.

Entry

This is an OU level 3 module. OU level 3 modules build on study skills and subject knowledge acquired from studies at levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with The Open University. You are expected to bring to the module some knowledge of:

• Calculus – definition of differentiation; ability to differentiate a variety of functions; Taylor's theorem with remainder; partial derivatives; understanding of continuity and convergence
• Matrices – ability to manipulate equations with matrices and vectors; Gaussian elimination; eigenvalues and eigenvectors; linear dependence and independence.

You could get the necessary background from one of our level 2 mathematics modules Pure mathematics (M208), Mathematical methods, models and modelling (MST210), Mathematical methods (MST224) or the discontinued module Mathematical methods and models (MST209), or equivalent. You are more likely to successfully complete this module if you have acquired your prerequisite knowledge through passing at least one of these recommended modules.

You can try our self-assessment diagnostic quiz to help you determine if you are adequately prepared for this module.

If you have any doubt about the suitability of the module, please speak to an adviser.

Preparatory work

If you would like to do some preparatory reading, you could choose from:

• E. W. Cheney, D. R. Kincaid (2008) Numerical Mathematics and Computing, Brooks Cole, ISBN 10: 0-495-11475-8
• R. L. Burden, J. D. Faires (2011) Numerical Analysis, Brooks Cole, ISBN 10: 0-538-73563-5

For an introduction to linear algebra:

• H. Anton, C. Rorres (2010) Elementary Linear Algebra: With Supplemental Applications, John Wiley & Sons, ISBN 978-0-470-56157-7

The following material from Pure mathematics (M208) would be very useful:

• Linear Algebra Block: Unit 2 Linear Equations and Matrices; Unit 3 Vector Spaces; Unit 5 Eigenvectors.
• Analysis Block A: Unit 2 Sequences; Unit 4 Continuity.
• Analysis Block B: Unit 1 Limits, Unit 2 Differentiation.

If you have a disability or additional requirement

You will need to spend considerable amounts of time using a personal computer.

Study materials

What's included

Module texts and website, including access to Maxima mathematical software which you need to download.

You will need

Scientific calculator, but not one that is designed or adapted to offer any of the following facilities: Algebraic manipulation, differentiation or integration, language translation or can communicate with other devices or the internet. It also cannot have retrievable information stored in it such as databanks, dictionaries, mathematical formulae or text..

We recommend you access the internet at least once a week during the module to download module resources and assignments, and to keep up to date with module news.

Computing requirements

You will need a device with internet access to study this module as a web browser is used to access learning materials and activities. Any other computer-based activities you will need to carry out, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment, are specified in the module materials. If any additional software is needed for these tasks it will either be provided or is freely available. You may need administrative privileges to install software required for this module. Windows 10 S is not suitable as it restricts software installation to software available in the Windows Application Store.

Suitable devices are:

• A Windows desktop or laptop running Windows 7 or later operating system
• A Macintosh desktop or laptop running OS X 10.8 or later operating system.

Some software will not run on Linux, iOS or Android devices.

A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. However, these devices may not be suitable for some activities. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop device that uses the Windows or OS X operating system in case you are unable to carry out all activities on your mobile device.

Recent versions of the following browsers are suitable for carrying out web-based activities:

• Safari
• Chrome
• Firefox
• Edge

Or Internet Explorer 9 and above.

Using a browser upgraded to the latest version will maximise security when accessing the internet.

Using company or library computers may prevent you accessing some internet materials or installing additional software.

To be able to talk and listen in our online discussions you will need both a microphone and speakers/headphones.

Devices with small screens may make it difficult to view the material provided and carry out the activities. However, a device that has a resolution of at least 1024 pixels horizontally and also at least 768 pixels vertically should be adequate.

See our Skills for OU study website for further information about computing skills for study and educational deals for buying Microsoft Office software.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.

Contact us if you want to know more about study with The Open University before you register.

Assessment

The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.

Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.