268 research outputs found
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
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