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Your search keyword '"Primary 46C05, 41A29, Secondary 41A65, 90C25, 90C26"' showing total 2 results
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2 results on '"Primary 46C05, 41A29, Secondary 41A65, 90C25, 90C26"'

1. On angles between convex sets in Hilbert spaces

Author

Bauschke, Heinz H., Ouyang, Hui, and Wang, Xianfu

Subjects
Mathematics - Optimization and Control, Primary 46C05, 41A29, Secondary 41A65, 90C25, 90C26
Abstract

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., Comment: 18 pages

Published
2020

2. On angles between convex sets in Hilbert spaces

Author

Hui Ouyang, Heinz H. Bauschke, and Xianfu Wang

Subjects
Pure mathematics, Generalization, Applied Mathematics, 010102 general mathematics, Hilbert space, Regular polygon, 01 natural sciences, Linear subspace, 010101 applied mathematics, symbols.namesake, Nonlinear system, Intersection, Optimization and Control (math.OC), Convergence (routing), FOS: Mathematics, symbols, Order (group theory), 0101 mathematics, Primary 46C05, 41A29, Secondary 41A65, 90C25, 90C26, Mathematics - Optimization and Control, Analysis, Mathematics
Abstract

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., 18 pages

Published
2020

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