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A new projection method for finding the closest point in the intersection of convex sets
In this paper we present a new iterative projection method for finding the
closest point in the intersection of convex sets to any arbitrary point in a
Hilbert space. This method, termed AAMR for averaged alternating modified
reflections, can be viewed as an adequate modification of the Douglas--Rachford
method that yields a solution to the best approximation problem. Under a
constraint qualification at the point of interest, we show strong convergence
of the method. In fact, the so-called strong CHIP fully characterizes the
convergence of the AAMR method for every point in the space. We report some
promising numerical experiments where we compare the performance of AAMR
against other projection methods for finding the closest point in the
intersection of pairs of finite dimensional subspaces
Exact Convergence Rates of Alternating Projections for Nontransversal Intersections
We consider the convergence rate of the alternating projection method for the
nontransversal intersection of a semialgebraic set and a linear subspace. For
such an intersection, the convergence rate is known as sublinear in the worst
case. We study the exact convergence rate for a given semialgebraic set and an
initial point, and investigate when the convergence rate is linear or
sublinear. As a consequence, we show that the exact rates are expressed by
multiplicities of the defining polynomials of the semialgebraic set, or related
power series in the case that the linear subspace is a line, and we also decide
the convergence rate for given data by using elimination theory. Our methods
are also applied to give upper bounds for the case that the linear subspace has
the dimension more than one. The upper bounds are shown to be tight by
obtaining exact convergence rates for a specific semialgebraic set, which
depend on the initial points.Comment: 22 pages, 1 figur
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