2020/08/21 by Heinz H. Bauschke, Bauschke, Heinz H., Hui Ouyang +3
Computer Science · Engineering · Mathematics · #41A29 #90C25 #90C26 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Primary 46C05 #Secondary 41A65 #Sparse and Compressive Sensing Techniques #math.OC #msc:41A29 #msc:41A65 #msc:46C05 #msc:90C25 #msc:90C26
paper · pdf · doi:10.48550/arxiv.2008.09313
18 pages
openalex publication_date 2020/08/21 · arxiv created 2021/05/06 · arxiv updated 2021/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of the angle between two subspaces has a long history, dating back to Friedrichs's work in 1937 and Dixmier's work on the minimal angle in 1949. In 2006, Deutsch and Hundal studied extensions to convex sets in order to analyze convergence rates for the cyclic projections algorithm. In this work, we characterize the positivity of the minimal angle between two convex cones. We show the existence of, and necessary conditions for, optimal solutions of minimal angle problems associated with two convex subsets as well. Moreover, we generalize a result by Deutsch on minimal angles from linear subspaces to cones. This generalization yields sufficient conditions for the closedness of the sum of two closed convex cones. This also relates to conditions proposed by Beutner and by Seeger and Sossa. Furthermore, we investigate the relation between the intersection of two cones (at least one of which is nonlinear) and the intersection of the polar and dual cones of the underlying cones. It turns out that the two angles involved cannot be positive simultaneously. Various examples illustrate the sharpness of our results.