Approximate solutions to variational inequality over the fixed point set of a strongly nonexpansive mapping
© Iemoto et al.; licensee Springer. 2014
Received: 3 September 2013
Accepted: 13 February 2014
Published: 25 February 2014
Variational inequality problems over fixed point sets of nonexpansive mappings include many practical problems in engineering and applied mathematics, and a number of iterative methods have been presented to solve them. In this paper, we discuss a variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping on a real Hilbert space. We then present an iterative algorithm, which uses the strongly nonexpansive mapping at each iteration, for solving the problem. We show that the algorithm potentially converges in the fixed point set faster than algorithms using firmly nonexpansive mappings. We also prove that, under certain assumptions, the algorithm with slowly diminishing step-size sequences converges to a solution to the problem in the sense of the weak topology of a Hilbert space. Numerical results demonstrate that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm.
MSC:47H06, 47J20, 47J25.
when is strongly monotone and Lipschitz continuous. Problem (2) contains many applications such as signal recovery problems , beam-forming problems , power-control problems [13, 14], bandwidth allocation problems [15–17], and optimal control problems . References [11, 19], and  presented acceleration methods for solving Problem (2) when A is strongly monotone and Lipschitz continuous. Algorithms were presented to solve Problem (2) when A is (strictly) monotone and Lipschitz continuous [15, 17]. When and is continuous (and is not necessarily monotone), a simple algorithm, (), was presented in  and the algorithm converges to a solution to Problem (2) under some conditions.
Algorithm (5) potentially converges in the fixed point set faster than algorithm (4). Here, we can see that the mapping, , satisfies the strong nonexpansivity condition , which is a weaker condition than firm nonexpansivity. This implies that the previous algorithms in [14, 21], which can be applied to Problem (2) when T is firmly nonexpansive, cannot solve Problem (2) when T is strongly nonexpansive.
In this paper, we present an iterative algorithm for solving the variational inequality problem with a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping and show that the algorithm weakly converges to a solution to the problem under certain assumptions.
The rest of the paper is organized as follows. Section 2 covers the mathematical preliminaries. Section 3 presents the algorithm for solving the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping, and its convergence analyses. Section 4 provides numerical comparisons of the algorithm with the previous algorithm in  and shows that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm. Section 5 concludes the paper.
Throughout this paper, we will denote the set of all positive integers by ℕ and the set of all real numbers by ℝ. Let H be a real Hilbert space with inner product and its induced norm . We denote the strong convergence and weak convergence of to by and , respectively. It is well known that H satisfies the following condition, called Opial’s condition : for any satisfying , holds for all with ; see also [5, 6, 24]. To prove our main theorems, we need the following lemma, which was proven in ; see also [5, 6, 26].
Lemma 2.1 ()
Assume that and are sequences of non-negative numbers such that for all . If , then exists.
2.1 Strong nonexpansivity and fixed point set
Let T be a mapping of H into itself. We denote the fixed point set of T by ; i.e., . A mapping is said to be nonexpansive if for all . is closed and convex when T is nonexpansive [5, 6, 24, 27]. is said to be strongly nonexpansive  if T is nonexpansive and if, for bounded sequences , implies . The following properties for strongly nonexpansive mappings were shown in :
is closed and convex when is strongly nonexpansive because T is also nonexpansive.
If are strongly nonexpansive, then ST is also strongly nonexpansive, and when .
If is strongly nonexpansive and if is nonexpansive, then is strongly nonexpansive for . If , then . In particular, since the identity mapping I is strongly nonexpansive, the mapping is strongly nonexpansive. Such U is said to be averaged nonexpansive.
Then T is strongly nonexpansive and .
Then S is nonexpansive [, Proposition 4.2] and . Hence, T is strongly nonexpansive and . is referred to as a generalized convex feasible set [10, 32] and is defined as the subset of that is closest to in the mean square sense. Even if , is well defined. holds when . Accordingly, is a generalization of .
A mapping is said to be firmly nonexpansive  if for all (see also [24, 27, 34]). Every firmly nonexpansive mapping F can be expressed as given some nonexpansive mapping T [24, 27, 34]. Hence, the class of averaged nonexpansive mappings includes the class of firmly nonexpansive mappings.
2.2 Variational inequality
We denote the solution set of the variational inequality problem by . The monotonicity and hemicontinuity of A imply that [, Subsection 7.1]. This means that is closed and convex. is nonempty when is monotone and hemicontinuous, and is nonempty, compact, and convex [, Theorem 7.1.8].
Example 2.3 Let be convex and continuously Fréchet differentiable and . Then A is monotone and hemicontinuous.
3 Optimization of variational inequality over fixed point set
In this section, we present an iterative algorithm for solving the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping and its convergence analyses. We assume that is a strongly nonexpansive mapping with and that is a monotone, hemicontinuous operator.
Step 0. Choose , , and arbitrarily, and let .
Step 2. Update , and go to Step 1.
To prove our main theorems, we need the following lemma.
, , and the existence of an satisfying .
Then is bounded.
From , the boundedness of , and Lemma 2.1, the limit of exists for all , which implies that is bounded.
Hence, the condition, , and Lemma 2.1 guarantee that the limit of exists for all . We thus conclude that is bounded. □
Now, we are in the position to perform the convergence analysis on Algorithm 3.1 under condition (A) in Lemma 3.1.
Then Algorithm 3.1 converges weakly to a point in .
Prove that and are bounded.
Prove that and hold.
Prove that converges weakly to a point in .
Choose arbitrarily. From the inequality, , and Lemma 3.1, we deduce that is bounded.
- (b)Put for any . Then, from , for any , we can choose such that , and for all . Also, there exists such that for all because of . Since , we have
- (c)From the boundedness of , there exists a subsequence of such that converges weakly to a point . From the nonexpansivity of T and (12), it is guaranteed that T is demiclosed (i.e., and imply ). Hence, we have . From (9), we get, for any and for any ,
This is a contradiction. Thus, . This implies that every subsequence of converges weakly to the same point in . Therefore, converges weakly to . This completes the proof. □
Remark 3.2 If the sequence satisfies the assumptions in Theorem 3.1, we need not assume that or that exists such that in condition (B) (see also [, Remark 7(c)]).
instead of in Algorithm 3.1. Since and V is bounded, is bounded. The Lipschitz continuity of A means that (), where L (>0) is a constant, and hence, is bounded. We can prove that Algorithm 3.1 with Equation (13) and and satisfying the conditions in Theorem 3.1 (or Theorem 3.2) weakly converges to a point in by referring to the proof of Theorem 3.1 (or Theorem 3.2).
We prove the following theorem under condition (B) in Lemma 3.1. The essential parts of a proof are similar those of Lemma 3.1 and Theorem 3.1, so we will only give an outline of the proof below.
If and if there exists such that , then the sequence converges weakly to a point in .
Prove that and are bounded.
Prove that holds.
Prove that converges weakly to a point in .
From Lemma 3.1, it follows that the limit of exists for all , and hence and are bounded.
- (b)Let and put . Since , the condition, , holds. As in the proof of Theorem 3.1(b), for any , there exists such that
Following the proof of Theorem 3.1(c), there exists a subsequence such that converges weakly to . Assume that another subsequence of converges weakly to w. Then we also have . Since the limit of exists for , Opial’s theorem  guarantees that . This implies that every subsequence of converges weakly to the same point in , and hence, converges weakly to . This completes the proof. □
As we mentioned in Section 1, to solve constrained optimization problems whose feasible set is the fixed point set of a nonexpansive mapping T, Algorithm 3.1 must converge in early in the execution. Therefore, it would be useful to use a large parameter α () when a strongly nonexpansive mapping is represented by . Theorem 3.1 has the following consequences.
Then converges weakly to a point in .
Proof Since every averaged nonexpansive mapping is strongly nonexpansive and for , Theorem 3.1 implies Corollary 3.1. □
By following the proof of Theorem 3.2 and Corollary 3.1, we get the following.
If and if there exists such that , then converges weakly to a point in .
4 Numerical examples
Let us apply Algorithm 3.1 and the algorithm in  to the following variational inequality problem.
where is positive semidefinite, , and (). Find .
We set Q as a diagonal matrix with diagonal components and choose (, ) to be Mersenne Twister pseudo-random numbers given by the random-real function of srfi-27, Gauche.a We also set and . The compiler used in this experiment was gcc.b The double-precision floating points were used for arithmetic processing of real numbers. The language was C.
The convergence of to 0 implies that algorithm (15) converges to a point in .
to check that algorithm (15) is stable.
We studied a variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping in a Hilbert space and devised an iterative algorithm for solving it. Our convergence analyses guarantee that the algorithm weakly converges to a solution under certain assumptions. We gave numerical results to support the convergence analyses on the algorithm. The results showed that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm.
We used the Gauche scheme shell, 0.9.3.3 [utf-8,pthreads], x86_64-apple-darwin12.4.1.
We used gcc version 4.2.1 (Based on Apple Inc. build 5658) (LLVM build 2336.11.00).
For example, we set a large parameter, i.e., much more than : .
We are sincerely grateful to the Lead Guest Editor, Qamrul Hasan Ansari, of Special Issue on Variational Analysis and Fixed Point Theory: Dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday, and the two anonymous referees for helping us improve the original manuscript. This work was supported by the Japan Society for the Promotion of Science through a Grant-in-Aid for JSPS Fellows (08J08592) and a Grant-in-Aid for Young Scientists (B) (23760077).
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