To give students a broad understanding of the theoretical and numerical aspects of non-linear optimisation

Criteria for Optimality. Conditions for linear equality- and inequality-constrained problems. First-order and second-order Karush-Kuhn-Tucker (KKT) conditions for general non-linearly constrained problems. Unconstrained Optimisation. Univariate Functions: Line Searches. Multivariate Functions: Steepest Descent and Newton's Method, Modifications of Newton's Method including Levenberg-Marquardt Method. Conjugate Gradient Methods. Constrained Optimisation. Penalty and Barrier Function Methods. Computational limitations of penalty function methods - ill-conditioning. Exact Penalty Function Methods. The module will include at least one computer-based project requiring students to select and implement a suitable algorithm for the solution of a non-trivial optimisation problem using either Fortran or Matlab.

Learning Outcome Programme Outcome(s) Programme Area(s) Assessment Mode(s) 1. Derive convergence conditions for unconstrained minimisation, use the concept of rate of convergence to analyse the steepest descent method. 1 1 Mid-term and end of semester assessment 2. Prove Zoutendijk's Theorem and use it to show that the Wolfe conditions guarantee convergence for a wide category of minimisation algorithms. 1 1 Mid-term and end of semester assessment 3. Derive and analyse gradient-based methods including Newton's method, the BFGS method and the various conjugate gradient methods. 1 1 Mid-term and end of semester assessment 4. Derive and analyse Trust Region methods including the Cauchy point method, the Dogleg method, Two-dimensional subspace method and the ôNearly Exactö method. 1 1 Mid-term and end of semester assessment 5. Prove the first and second-order (KKT) necessary conditions for equality- and inequality-constrained minimisation problems and use them to analyse algebraically simple problems. 1 1 Mid-term and end of semester assessment 6. Derive the various methods for the solution of equality-constrained Quadratic Programs (QP's), show how they can be extended to inequality-constrained QP's and use them to solve some simple QP's. 1 1 Mid-term and end of semester assessment 7. Write Matlab code to implement one of the methods mentioned above and apply the code to a non-trivial optimisation problem. 1 1 Mid-term PC-based project.

None.

None at all.

Conventional lectures with slides in Adobe Acrobat format displayed with data projector. Lecture notes will be available via the internet and in printed form. Tutorials will provide students with assistance in solving set problems and an opportunity for further interaction including help with computer-based problem-solving.

As is appropriate for a final year module, current work in non-linear optimisation is incorporated in the course material and updated yearly.