An iterative algorithm for approximating convex minimization problem
Introduction
The problem of finding an element in the intersection of the fixed point sets of a finite family of nonexpansive mappings has attracted much attention because of its extraordinary utility and broad applicability in many branches of mathematical science and engineering. For example, if the nonexpansive mappings are projection onto some closed convex sets Ci(i ∈ I) in a real Hilbert space H, then such a fixed point problem becomes the convex feasibility problem of finding a point in ⋂i∈ICi (see [1], [2], [3]). Moreover, the problem of finding an optimal point that minimizes a given cost function Θ : H → R over the intersection is of broad interdisciplinary interest due to its practical importance (please see [3], [4], [5], [6], [7]). In particular, a simple algorithmic solution to the problem of minimizing a quadratic function over the intersection has been desired in many applications including the set theoretic signal estimation which we refer to [8], [9].
Let H be a real Hilbert space, and A a bounded linear operator on H. Throughout this paper, we always assume that A is strongly positive; that is, there is a constant γ > 0 such thatRecall that a mapping T of H into itself is called nonexpansive iffor all x, y ∈ H.
Let Ti, where i = 1, 2, …, an infinite family of nonexpansive self-mappings of H. Let Fix(Ti) denote the fixed point set of Ti, i.e., Fix(Ti) := {x ∈ H : Tix = x}, and let .
Concerning a finite family of nonexpansive mappings , the following iterative algorithmwhere Tn = Tn mod N and the mod function takes values in {1, 2, … , N} for approximating the unique minimizer x∗ of the quadratic function θwhere has been considered by many authors. In particular, in 1998, Yamada et al. [10] proved that the iterative algorithm (1) converges to the unique minimizer of θ over provided {Tn} satisfyand {λn} is a sequence in (0, 1) satisfying the following control conditions:
- (C1)
limn→∞λn = 0;
- (C2)
;
- (C3)
.
Subsequently, in 2003, Xu [11] obtained a complementary result to Theorem 3 of Yamada et al. [10] by replacing (C3) with the general condition (C3′): limn→∞λn/λn+N = 1. We note that condition (C3) and (C3′) are not comparable in general. See [11] for more details.
All of the above bring us the following conjectures? Question 1.1 Let Ti : H → H be nonexpansive and A a bounded linear operator on H. Could we remove the commutativity assumption (3) imposed on mappings ? Could we weaken or remove the control condition (C3) on parameter {λn}? Could we construct an iterative algorithm of an infinite family of nonexpansive mappings ? Could we construct an iterative algorithm to approximate the solution of the following quadratic minimization problem:where ? It is our purpose in this paper that we suggest and analyze an iterative algorithm without the assumption of any type of commutativity on an infinite family of nonexpansive mappings as follows:
Algorithm 1.1
Let T1, T2, … be an infinite family of mappings of H into itself and let α1, α2, … be real numbers such that 0 ⩽ αi ⩽ 1 for every i ∈ N. For any n ∈ N, we define a mapping Wn of H into itself as follows:Such a mapping Wn is called the W-mapping generated by Tn, Tn−1, … , T1 and αn, αn−1, … , α1.
For any given x0 ∈ H, the sequence {xn} can be generated iteratively bywhere {λn} is a sequence in (0, 1), β is a constant in (0, 1), u ∈ H is arbitrarily fixed point, A is a strongly positive linear bounded operator with coefficient γ > 0 and Wn is the W-mapping defined by (5).
We show that the iterative algorithm (6) converges to the unique minimizer of the quadratic function (2) over the common fixed point sets F of an infinite family of nonexpansive mappings under the control conditions (C1) and (C2) on parameter {λn}. Our result answer all questions in Question 1.1 and extend as well as improve the corresponding results announced by many authors.
Section snippets
Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
Let B(H) denotes the set of all bounded linear operators from H to H. The following fact characterizes the minimizer of a quadratic function θ. It can be found in [12]. Lemma 2.1 Let H be a Hilbert space, b ∈ H and A ∈ B(H) a self-adjoint and strongly positive operator. Let θ be defined by (2). Then for a nonempty closed and convex subset C of H, there exists a unique minimizer x∗ of θ over C; x∗ ∈ C is a[12]
Main result
Now we study the convergence of iterative algorithm (6). Theorem 3.1 Let H be a real Hilbert space, b ∈ H, A ∈ B(H) a self-adjoint, strongly positive operator and an infinite family of nonexpansive mappings from H to H such that the common fixed points set . Suppose the control conditions (C1) and (C2) on parameters {λn} are satisfied. Then for any given x0 ∈ H, the sequence {xn} generated by (6) converges strongly to the unique minimizer x∗ of the quadratic function θ of (2) over F. Proof The
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