Accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes

The von Neumann-Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas: Firstly, each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection, and secondly, it is easy to project onto a finite intersection of such hyperplanes. We give conditions for which our accelerations converge strongly. Finally, we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem.Comment: 16 pages, 3 figures (Corrected minor typos in Remark 2.2, Algorithm 2.5, proof of Theorem 3.12, as well as elaborated on certain proof

Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finally, we discuss the relationship between the robust form of generalization differentiation and its one sided counterpart.Comment: This submission corrects errors from the previous version after referees' comments. Changes are in Proposition 2.4, Proposition 4.12, and Sections 7 and