How to project onto the intersection of a closed affine subspace and a hyperplane

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Germany), 2024-10, Vol.100 (2), p.411-435
• Main Authors: Bauschke, Heinz H., Mao, Dayou, Moursi, Walaa M.
• Format: Article
• Language: English
• Subjects: Business and Management; Calculus of Variations and Optimal Control; Optimization; Computed tomography; Hyperplanes; Image reconstruction; Intersections; Linear algebra; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; OR for the classroom; Subspaces
• ISSN: 1432-2994; 1432-5217
• DOI: 10.1007/s00186-024-00866-z

Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint.