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Best approximation mappings in Hilbert spaces

Abstract

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of a BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspaces to nonempty closed convex sets and from \({\mathbb {R}}^{n}\) to general Hilbert spaces. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al. proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in \({\mathbb {R}}^{n}\). We generalize their result from \({\mathbb {R}}^{n}\) to general Hilbert spaces and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAMs with circumcenter mappings in Hilbert spaces and we report on Cegielski’s results on relationships with strongly quasinonexpansive mappings.

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Acknowledgements

HHB and XW were partially supported by NSERC Discovery Grants. The authors thank Andrzej Cegielski, the referees, and the editor for comments that were very helpful and constructive.

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Correspondence to Xianfu Wang.

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Bauschke, H.H., Ouyang, H. & Wang, X. Best approximation mappings in Hilbert spaces. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01718-y

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Keywords

  • Best approximation mapping
  • Linear convergence
  • Fixed point set
  • Best approximation problem
  • Projector
  • Circumcentered isometry method
  • Circumcentered reflection method
  • Linearly regular mappings
  • Method of alternating projections
  • Strongly quasinonexpansive mapping

Mathematics Subject Classification

  • Primary 90C25
  • 41A50
  • 65B99
  • Secondary 46B04
  • 41A65