An Iterative Algorithm of Solution for Quadratic Minimization Problem in Hilbert Spaces

Volume 2010, Article ID 717341,6pages doi:10.1155/2010/717341

Research Article

An Iterative Algorithm of Solution for Quadratic

Minimization Problem in Hilbert Spaces

Li Liu,

_{Guanghui Gu,}

_{and Yongfu Su}

1_{Department of Mathematics, Cangzhou Normal University, Hebei, Cangzhou 061001, China} 2_{Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China}

Correspondence should be addressed to Yongfu Su,[email protected]

Received 24 October 2009; Accepted 28 December 2009

Academic Editor: Jong Kim

Copyrightq2010 Li Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to introduce an iterative algorithm for finding a solution of quadratic minimization problem in the set of fixed points of a nonexpansive mapping and to prove a strong convergence theorem of the solution for quadratic minimization problem. The result of this article improved and extended the result of G. Marino and H. K. Xu and some others.

1. Introduction and Preliminaries

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,1,2and the references therein. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping on a real Hilbert spaceH:

min x∈C

1

2Ax, x − x, u, 1.1

whereCis the fixed point set of a nonexpansive mappingT defined onH, anduis a given point inH. LetAbe a strongly positive operator defined onH, that is, there is a constant

γ >0 with the property

Ax, x ≥γx2_{, ∀x∈H. 1.2}

Then minimization1.1has a unique solutionx∗∈Cwhich satisfies the optimality condition

In1,2it is proved that the sequence{xn}generated by the following algorithm

xn1 I−αnATxnαnu, n≥0 1.4

converges in norm to the solutionx∗of1.1provided that the sequence{αn}in0,1satisfies conditions

lim

n→ ∞αn0, C1

∞

n1

αn∞, C2

and additionally, either the condition

∞

n1

|αn1−αn|<∞, C3

or the condition

lim n→ ∞

|αn1−αn|

αn1

0. C4

The purpose of this paper is to introduce the following iterative algorithm:

xn1 I−αnAynαnu,

yn βnxn1−βnTxn,

1.5

and to prove that the iterative sequence {xn} defined by 1.5 converges strongly to the solutionx∗of1.1under the conditionsC1,C2and 0< a≤βn≤b <1 for some constants

a, b.

Lemma 1.1see3,4. Let{xn}and{yn}be bounded sequences in a Banach spaceXsuch that

xn1λnxn 1−λnyn, n≥0, 1.6

where{λ_n}is a sequence in0,1such that

0<lim inf

n→ ∞ λn≤lim sup_{n→ ∞} λn<1. 1.7

Assume that

lim sup n→ ∞

_{yn1−yn− xn1−xn}

≤0. _1.8

Lemma 1.2see1. Assume thatAis a strongly positive linear bounded operator on a real Hilbert spaceHwith coefficientγ >0and0< α≤ A−1_{. ThenI−αA ≤1−αγ.}

Lemma 1.3see5. LetH be a Hilbert space,K a closed convex subset ofH, andT :K → K

a nonexpansive mapping with nonempty fixed point set FT. If{xn} is a sequence in K weakly

converging toxand ifx_n−Tx_nconverges strongly to0, thenxTx.

Lemma 1.4see6. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤

1−γnanγnδn, 1.9

where{γn}is a sequence in0,1and{δn}is a real sequence such that

i∞_n1γn∞,

iilim sup_{n→ ∞}δ_n≤0or∞_n1γ_n|δ_n|<∞.

Thenlim_{n→ ∞}a_n0.

2. Main Results

Theorem 2.1. Suppose thatAis strongly positive operator with coefficientγ > 0as given in1.2. Suppose that the sequences{α_n},{β_n}satisfy the conditionsC1,C2and0 < a≤βn ≤b <1for

some constantsa, b. Then the sequence{x_n}generated by algorithm1.5converges strongly to the unique solutionx∗of the minimization problem1.1.

Proof. First we show that {x_n} is bounded. As a matter of fact, take p ∈ FT and use

Lemma 1.2to obtain

_{xn1−pI−αnA}

βnxn1−βnTxn−pαnu−Ap

≤1−γαnxn−pαnu−Ap.

2.1

By induction we can get

_{xn−p≤maxx0−p,}1

γu−Ap

, n≥0. 2.2

Hence,{x_n}is bounded and so is{y_n}. Next rewritex_n1in the form

xn1 1−λnxnλnzn, 2.3

where

λn1−1−αnβn, 2.4

zn α_λnβn

n I−Axn 1−βn

λn I−αnATxn

αn

Sinceαn → 0 and 0< a≤βn≤b <1, then

0<lim inf

n→ ∞ λn≤lim sup_{n→ ∞} λn<1. 2.6

Next some manipulations give us that

zn1−zn

βn1αn1

λn1 I−Axn1−

βnαn

λn I−Axn

αn1

λn1 −

αn

λn

u

1−βn1

λn1 Txn1−Txn−

1−βn1

αn1

λn1 ATxn1−Txn

1−βn1

λn1 −

1−βn

λn

Txn− α_λn1 n1 −

αn

λn

1−β_nATx_n

−αn1

λn1

βn−βn1

ATxn.

2.7

Therefore,

zn1−zn − xn1−xn ≤ βn_λ1αn1

n1 I−Axn1

βnαn

λn I−Axn

αn1

λn1 −

αn

λn

u

1−βn1

λn1 −

1

xn1−xn

1−βn1

αn1

λn1 Axn1−xn

1−βn1

λn1 −

1−βn

λn

Txnα_λn1 n1 −

αn

λn

1−βnATxn

αn1

λn1

_{βn−βn1ATxn.}

2.8

Sinceλ_n1−1−α_nβ_nandα_n → 0, then

lim n→ ∞

1−βn

λn nlim→ ∞ 1−

αnβn

λn

1. 2.9

Then last inequality implies that

lim sup

n→ ∞ zn1−zn − xn1−xn≤0, 2.10

and so an application ofLemma 1.1asserts that

lim

By2.5we have that

zn−Txn α_λnβn

n I−Axn

1−β_n

λn −1

Txn−

1−βnαn

λn ATxn

αn

λnu. 2.12

Again sinceα_n → 0,{x_n}is bounded, andλ_n 1−1−α_nβ_n, then we deduce from2.12

that

lim

n→ ∞zn−Txn0. 2.13

This together with2.11yields

lim

n→ ∞xn−Txn0. 2.14

By usingLemma 1.3, we obtainω_wx_n⊂FT, whereω_wx_n {z:∃ x_nk z}is the set of weakω-limit points of sequence{xn}.

Let x∗ be the unique solution to the minimization 1.1. Then by the definition of algorithm1.5, we can write

xn1−x∗ I−αnAβnxn1−βnTxn−x∗αnu−Ax∗. 2.15

SinceHis a Hilbert space, then we have that

xn1−x∗2≤I−αnAβnxn1−βnTxn−x∗22αnu−Ax∗, xn1−x∗

≤1−γα_nx_n−x∗2α_nu−Ax∗, x_n1−x∗.

2.16

However, we can take a subsequence{xnk}of{xn}such that

lim sup

n→ ∞ u−Ax ∗_{, x}

n−x∗ lim

k→ ∞u−Ax ∗_{, x}

nk−x∗, 2.17

and also {xnk} converges weakly to a fixed point p ∈ FT. It follows from optimality

condition1.3that

lim sup

n→ ∞ u−Ax ∗_{, x}

n−x∗u−Ax∗, p−x∗≤0. 2.18

References

1 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”

Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.

2 H.-K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.

3 T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,”Fixed Point Theory and Applications, vol. 2005, no. 1, pp. 103–123, 2005.

4 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,”Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

5 K. Goebel and W. A. Kirk,Topics in Metric Fixed Point Theory, vol. 28 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.