PhD Defence: The Pareto Eigenvalue Problem

CANDIDATE: Thomas Kielstra

ABSTRACT: The Pareto eigenvalue problem has attracted the attention of researchers from a variety of different backgrounds. While it can be stated easily, it has proven to be challenging to determine some of its basic properties. Since the Pareto eigenvalue problem was first introduced, many researchers have been interested in determining the Pareto capacity, which is the maximum number of Pareto eigenvalues a real n × n matrix can have. In this thesis, we analyse the Pareto eigenvalue problem from varying approaches, utilizing theory from different areas of mathematics. We explore the use of semialgebraic geometry for general linear algebra. Then we use semialgebraic geometry to gain a better understanding of the properties of Pareto eigenvalues. We define a semialgebraic set of the real n × n matrices, called the principally simple matrices, where all matrices in a connected component of the principally simple matrices have the same number of Pareto eigenvalues. Using semialgebraic geometry, we show that the Pareto capacity is, in theory, computable but not currently tractable. The Pareto capacity is only known for n = 1, 2 and 3. We show how perturbation theory and exhaustive computational searches could be applied to the Pareto eigenvalue problem. Combining these techniques, we are able to show that a principally simple matrix with the maximum number of Pareto eigenvalues for n = 4 exists. Additionally, we decrease the upper bound for the Pareto capacity for n⩾ 5. Many researchers have developed numerical methods to try and find a Pareto eigenvalue for any given matrix. We attempt to improve some of these methods by using population-based optimization algorithms.