Research

Stochastic programming has become a common tool to study and model decision problems with the presence of uncertainty. These models are usually based on the use of multivariate probability distributions describing the uncertainty in the input data. The exact or approximating methods that are important for applications mainly deal with discrete empirical probability distributions that are described by a list of realizations (called scenarios) and related probabilities.

In most applications, the multivariate distributions do not come in a form suitable for the optimization model, being either continuous, discrete with too many data points, or specified by a set of statistical properties. Hence, to use a stochastic programming model, one has to transform the given distribution to scenarios - a process known as scenario generation.

An important question is: How can we come up with a good discretization of the multivariate distributions that trades off the following two major points?

1. We want the discretization to be such that solving the stochastic programming problem produces a solution which is abritrarily close to what we would have obtained if we were able to use the exact multivariate distributions.
2. For stochastic programming models to be solvable for any reasonably sized problem, the number if atoms in the discretized distribution distribution must be very low.

There have been many approaches taken to scenario generation, each with different strenghts and weaknesses. These include using sampling methods to provide a discretization, or starting with discretizations consisting of a small number of atoms and then iteratively using the solution in this case to work figure out where the discretization needs to be finer.

Projects

Mathematical Programming through Cones (2011) - In a mathematical programme one tries to minimise (or maximise) of a function given a set of constraints. Conic programming is the special case of this where one is optimising the function over a conic section. Conic programming covers a surprisingly wide class of problems and efficient solution algorithms are allowing one to solve previously intractible problems. This report outlines the basic theory of conic programmes before presenting some of its applications. Written under the supervision of Prof. Adam Letchford.

Particle Filters (2011) - Particles filters are used to make inference about underlying states based on noisy observations. This poster introduces two basic particle filters and applies them to a tracking problem.

OR against the U-boat (2011) - Operational Research came to prominence during the second world war where it was used to great effect by the allies. This website, written with Tim Park and Bethan Turner, describes the work Coastal Command Operational Research Section did against the persistant threat of German U-boats.

A Summary of Texture Classification Literature using Curvelets (2010) - This report gives a basic introduction to Curvelets before reviewing a body of available literature pertinent to texture analysis. Written under the supervision of Dr. Idris Eckley.

An Introduction to Tropical Hypersurfaces (2010) - Tropical Geometry is an emerging branch of algebraic geometry with promising applications. This dissertation, written under the supervision of Dr. Diane Maclagan, gives three different formulations of tropical hypersurfaces, and culminates in a geometric theorem which vastly simplifies the calculation of certain tropical hypersurfaces.

Home

Welcome. I am currently a PhD student at the new Statistics and Operational Research DTC based at Lancaster University. The STOR-i Doctoral Training Centre is a joint venture between the departments of Mathematics and Statistics and Management Science, and offers a four-year PhD programme developed and delivered with industrial partners. Prior to joining STOR-i I received a masters degree in Mathematics from Warwick University.

A major component of the STOR-i MRes is the completion of individual and group projects. Details of work I have completed during this year can be found in the Projects tab along with some other previous undertakings.

My PhD will focus on "Scenario Tree Generation" in Stochastic Programming and will be jointly supervised by Prof. Stein Wallace and Dr. Amanda Turner. For more details on this area, click on the Reseach tab.