Unit name | Optimisation |
---|---|

Unit code | MATH30017 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Tadic |

Open unit status | Not open |

Pre-requisites |
Analysis 1, Calculus 1, Linear Algebra and Geometry. Multivariable Calculus is desirable. |

Co-requisites |
none |

School/department | School of Mathematics |

Faculty | Faculty of Science |

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 (E.g., 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.

The proposed 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 would 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 would 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.

The aim the proposed unit is to make students acquainted with the main concepts, ideas, methods, tools and techniques of the mathematical optimisation. In particular, the unit should provide students with good understanding of the theoretical and numeral aspects of mathematical optimisation (described in (i) – (iv) and (a), (b) ).

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.

Lectures, problem sheets and office hours.

Formative assessment:

- weekly problem sheets

Summative assessment:

- One 2.5h exams (80%)
- Two computationally oriented homeworks (each 10%)

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

Further reading and references are available at http://www.maths.bristol.ac.uk/study/undergrad/current_units/index.html