Optimization Hierarchies For Distance-Avoiding Sets In Compact Spaces
Bram Bekker (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Olga Kuryatnikova ( Erasmus Universiteit Rotterdam)
Fernando Mário de Oliveira Filho (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Juan C. Vera (Tilburg University)
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Abstract
Witsenhausen’s problem asks for the maximum fraction αn of the (n − 1)-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. The best upper bounds for αn are given by extensions of the Lovász theta number. In this paper, optimization hierarchies based on the Lovász theta number, like the Lasserre hierarchy, are extended to Witsenhausen’s problem and similar problems. These hierarchies are shown to converge and are used to compute the best upper bounds for αn in low dimensions.