Academic advisor: Anders Forsgren
Sponsor: Swedish Research Council (VR)
The proposed project concerns further development of methods for solving large-scale nonlinear optimization problems. Our research on radiation therapy optimization has made us rethink the traditional approach for certain classes of optimization problems. Although some of the problems posed there have a large number of variables, they are ill-conditioned in the sense that there is a low-dimensional ``essential component'' of the problem and the high dimensionality comes into the ``fine-tuning'' of the solution to reach optimality. In our case, a quasi-Newton method was able to take advantage of this structure by generating iterates along a desirable path in the sense that it gave a solution with ``near-optimal'' objective function value in a low number of iterations. It was not enough for us to take the problem and give the optimal solution. Rather, it was necessary to take into account the path to the solution taken by the iterates.
This way of having a method that generates iterates along a desirable path is very interesting from an optimization perspective. We intend to deepen our knowledge in this research area. For our current problems, the behavior has been tied into properties of the linear conjugate-gradient method when applied to a linear equation, where the symmetric positive-definite matrix is highly ill-conditioned with few dominating eigenvalues. We intend to deepen this understanding, and also identify new areas where such techniques may be applicable. A very interesting line of research would be to investigate how such techniques might be applicable for identifying essential data in a data set of high dimension.
We also have more applied projects towards optimization of radiation therapy (with RaySearch Laboratories) and optimization of telecommunications networks (with Ericsson). There is a strong interplay between the applications and the fundamental research. The abovementioned regularizing behavior was an outcome of the research on radiation therapy. The research direction of the graduate student position in question here is towards fundamental method research, but there will be close connections to more applied areas. Numerical linear algebra will be a major component in the research, as efficient solution of particular systems of linear equations are of utmost importance in this research.