UBC Theses and Dissertations
Circumcenter operators in Hilbert spaces Hui, Ouyang
The best approximation problem is of central importance in convex optimization. It is popular to use the Douglas--Rachford splitting method or the method of alternating projections to solve this problem. In this thesis, we use a classical concept, circumcenter, in Euclidean geometry to solve the best approximation problem. First, we introduce the new notion, circumcenter operator. Symmetrical and asymmetrical formulae of the circumcenter operator are provided. A sufficient condition of the existence of the circumcenter is provided. A characterization of the existence of the circumcenter of three distinct points is presented. In addition, we define the new concept: circumcenter mapping induced by operators. When we choose the operators from sets of compositions of reflectors, we obtain the circumcenter mapping induced by reflectors, which is proper, i.e., the value of the circumcenter mapping induced by reflectors is always a unique point in the space. In light of this consequence, we are able to deduce the circumcenter method induced by reflectors. We also consider the circumcenter operator induced by projectors. Both proper and improper examples are provided. Moreover, we prove that for some special sets, the circumcenter methods induced by reflectors converge at least as fast as the MAP, symmetrical MAP or some of their accelerated version to solve the best approximation problem. We also find some drawbacks of the circumcenter mapping induced by reflectors. Finally, numerical experiments are implemented to compare convergence rates of seven solvers: four circumcenter methods induced by reflectors, DRM, MAP and symmetrical MAP. As the plots of performance profiles illustrate, the experimental results are consistent with our theoretical results in the thesis. Additional comparisons with the DRM and the circumcenter method induced by reflectors are made.
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