Quadratic optimization of fixed points for a family of nonexpansive mappings in Hilbert space
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Given a finite family of nonexpansive self-mappings of a Hilbert space, a particular quadratic functional, and a strongly positive selfadjoint bounded linear operator, Yamada et al. defined an iteration scheme which converges to the unique minimizer of the quadratic functional over the common fixed point set of the mappings. In order to obtain their result, they needed to assume that the maps satisfy a commutative type condition. In this paper, we establish their conclusion without the assumption of any type of commutativity. Finding an optimal point in the intersection F of the fixed point sets of a family of nonexpansive maps is one that occurs frequently in various areas of mathematical sciences and engineering. For example, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive maps. (See, e.g., [3, 4].) The problem of finding an optimal point that minimizes a given cost function Θ : Ᏼ → R over F is of wide interdisciplinary interest and practical importance. (See, e.g., [2, 4, 5, 7, 14].) A simple algorithmic solution to the problem of minimizing a quadratic function over F is of extreme value in many applications including the set-theoretic signal estimation. (See, e.g., [5, 6, 10, 14].) The best approximation problem of finding the projection PF (a) (in the norm induced by the inner product of Ᏼ) from any given point a in Ᏼ is the simplest case of our problem. Some papers dealing with this best approximation problem are [2, 9, 11]. Let Ᏼ be a Hilbert space, C a closed convex subset of Ᏼ, and Ti , where i = 1,2,...,N, a finite family of nonexpansive self-maps of C, with F := ∩ni=1 Fix(Ti ) = ∅. Define a quadratic function Θ : Ᏼ → R by 1 Θ(u) := Au,u − b,u 2
∀u ∈ Ᏼ,
(1)
where b ∈ Ᏼ and A is a selfadjoint strongly positive operator. We will also assume that B := I − A satisfies B < 1, although this is not restrictive, since µA is strongly positive Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 135–147 2000 Mathematics Subject Classification: 47H10 URL: http://dx.doi.org/10.1155/S1687182004309046
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Quadratic optimization
with I − µA < 1 for any µ ∈ (0,2/ A ), and minimizing Θ(u): = (1/2)µAu,u − µb,u over F is equivalent to the original minimization problem. Yamada et al. [13] show that there exists a unique minimizer u∗ of Θ over C if and only if u∗ satisfies
Au∗ − b,u − u∗ ≥ 0 ∀u ∈ C.
(2)
In their solution of this problem, Yamada et al. [13] add the restriction that the Ti satisfy
Fix TN · · · T1 = Fix T1 TN · · · T3 T2 = Fix TN −1 TN −2 · · · T1 TN .
(3)
There are many nonexpansive maps, with a common fixed point set, that do not satisfy (3). For example, if X = [0,1] and T1 and T2 are defined by T1 x = x/2 + 1/4 and T2 x = 3x/4, then Fix(T1 ,T2 ) = {2/5}, whereas Fix(T2 ,T1 ) = {3/10}. In our solution, we are able to remove restriction (3). We will take advantage of the modified Wittmann iteration scheme developed by Atsushiba
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