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

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. r 2004 Elsevier Inc. All rights reserved.

High-precision computation: Mathematical physics and dynamics

“…The consequent theory of this and related iterations is well understood in the convex case [21,22,23]. In the non-convex case the iteration, also called "divide-and-concur" [39], has been very successful in a variety of reconstruction problems (such as protein folding, 3SAT, spin glasses, giant Sudoku puzzles, etc.).…”

On the linear convergence of the circumcentered-reflection method

“…Among them, we are particularly interested in convex feasibility problems as in [1] and general convex inclusions (e.g. [19][20][21]). Simple examples, however, show that the circumcenter C(x) may not be defined for general convex sets at some "pathological" points x.…”

Techniques of Variational Analysis

“…Such methods are most useful when applied to feasibility problems whose constraint sets have more easily computable reflections and projections than does the intersection. When the underlying constraint sets are all convex, Douglas-Rachford methods are relatively well well-understood [6,12,11,7] -their behaviour can be analysed using nonexpansivity properties of convex projections and reflections. In the absence of convexity, recent result have assumed the constraint sets to possess other structural and regularity properties [10,1,20].…”

“…When N = 2, a useful surrogate is a pair of points, one from each set, which minimize the distance between the sets -a best approximation pair [6].…”