PhD ceremony Mr. P.J.C. Dickinson: The copositive cone, the completely positive cone and their generalisations

PhD ceremony: Mr. P.J.C. Dickinson, 16.15 uur, Academiegebouw, Broerstraat 5, Groningen

Dissertation: The copositive cone, the completely positive cone and their generalisations

Promotor(s): prof. M. Dür

Faculty: Mathematics and Natural Sciences

Often in life we are aiming for the best, whether this means being the fastest, the strongest, the most efficient or the most accurate. Mathematical Optimisation is a major tool in enabling us to make the best choices.

When using mathematical optimisation we need a model. Traditionally we have used linear models. During the last century, there were great successes connected to this approach; however, for many applications a linear model is insufficient, and instead we need a quadratic model. Unfortunately, optimisation for quadratic models is much more difficult due to nonconvexity creating local optima.

Copositive Optimisation provides a new way of providing exact reformulations of a large class of such problems. Vital to copositive optimisation is the Copositive Cone, and all of the difficulty from the original problems effectively goes into this cone.

During my PhD studies I have done research into fundamental properties of the copositive cone and its generalisations. This research has fallen into three main categories:

1. Applications for generalisations of copositive optimisation, along with related complexity results;

2. Geometric properties of the copositive cone and a related cone called the completely positive cone;

3. Approximations of the copositive cone and its generalisations.

This research has contributed to a significant improvement in our understanding of copositive optimization.