Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

  • Authors:

  • Heinz H. Bauschke;Patrick L. Combettes;D. Russell Luke

  • Affiliations:

  • Department of Mathematics and Statistics, University of Guelph, Guelph, Ont., Canada N1G 2W1;Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie -- Paris 6, 75005 Paris, France;The Pacific Institute for the Mathematical Sciences, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2004

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Abstract

We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses.