Undergraduate

Optimization

On this page

• What you will study
• Entry
• If you have a disability or additional requirement
• Study materials
• Teaching and assessment
• Students also studied
• Future availability

This course will interest you if you need to create mathematical models or if you use numerical software in industry, science, commerce or research. It’s concerned with the skills needed to represent real optimization problems as mathematical models, and with techniques used in numerical analysis and operational research for solving these models by computer. Explaining how and when modelling and numerical techniques can be applied, the course covers 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 minimisation problems. Knowledge of calculus and matrices is assumed.

Course facts

Register for the course

This course is available for study in the countries shown. Fees and financial support may vary by country.

What you will study

The course 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. About a quarter of your study time will be devoted to practical work. Computer programming is not part of the course.

In the broad area of operational research, the course 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 course 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 course 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 course 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 a Level 3 course. Level 3 courses 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 course some knowledge of:

• Calculus Definition of differentiation and integration; ability to differentiate and integrate 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 our Level 2 mathematics courses Pure mathematics (M208), or Mathematical methods and models (MST209), or the equivalent. Students are more likely to successfully complete this course if they have acquired their prerequisite knowledge through passing these recommended OU courses.

If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.

Preparatory work

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

• E. W. Cheney, D. R. Kincaid (2003) Numerical Mathematics and Computing, Brooks Cole, ISBN 0 534 38993 7
• R. L. Burden, J. D. Faires (2000) Numerical Analysis, Brooks Cole, ISBN 0 534 38216 9

For an introduction to linear algebra:

• H. Anton (2005) Elementary Linear Algebra: With Applications, John Wiley & Sons, ISBN 0 471 44902 4

The following material from Open University courses would be very useful:

• From Pure mathematics (M208):
  • 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.
• From Mathematical methods and models (MST209):
  • Unit 4 Vector Algebra
  • Unit 9 Matrices and Determinants
  • Unit 12 Functions of Several Variables
  • Unit 10 Eigenvalues and Eigenvectors
  • Unit 24 Vector Calculus
  • Unit 26 Numerical Methods for Differential Equations.

Your regional or national centre will be able to tell you where you can see reference copies, or you can buy selected materials from Open University Worldwide Ltd.

If you have a disability or additional requirement

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

If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.

You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.

Study materials

What's included

Main study texts, course guide, Mathcad 2001i and multimedia packages supplied on CD-ROM, website.

You will need

Scientific calculator.

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

Computing requirements

This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave. Some of your course software will be provided on disk.

You will need internet access and a computer. If you have purchased a new Windows computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual. Please note that you cannot use an Apple Mac or Linux computer unless it is running Windows using Boot Camp or similar dual-boot system.

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 course. 

Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

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

Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Students also studied

Students who studied this course also studied at some time:

• Graphs, networks and design (MT365)
• Mathematical methods and models (MST209)
• Applications of probability (M343)
• Mathematical modelling (MSXR209)
• Pure mathematics (M208)

Future availability

The details given here are for the course that starts in October 2012. We expect it to be available once a year.

Fees 2012

See fees and funding options for study from September 2012.

Study explained

Financial support

- find out if you qualify for support with your fees with our eligibility checker.

Study explained

- all you need to know about distance learning with the OU.

Student Reviews

One course has to be the least enjoyable and, for me, I am afraid it was M373. The only letter ...
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This was a thoroughly testing course that I strongly recommend as a Level 3 module for anyone doing a maths ...
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