Publications

We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of [omega]-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c. decompositions and propose some preprocessing ideas based on the spectral d.c. decomposition. We report on first numerical experience with this procedure which are very promising.

This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space R^m this can be done componentwise. That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.

Automatic differentiation is an often superior alternative to numerical differentiation that is yet unregarded for calculating derivatives in the optimization of imaging optical systems. We show that it is between 8% and 34% faster than numerical differentiation with central difference when optimizing various optical systems.

Freeform optical surfaces offer additional degrees of freedom for designing imaging systems without rotational symmetry. This allows for a reduction in the number of optical elements, leading to more compact and lightweight systems, while at the same time improving the image quality. This also enables new areas of application. Commonly used representations for freeform surfaces are x-y-polynomials, Zernike polynomials and NURBS. Radial basis functions (RBF) have been used for many years e.g. in artificial neural networks and functional approximation and can also be used to describe optical surfaces. In this contribution we investigate properties specific to RBF-based optical surfaces and compare the performance of RBF-based surfaces to other representations in selected optical imaging systems. Interesting aspects include the dependency on the number of RBF that are summed to form the surface, the locality structure and its effects on optimization.