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The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities
We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.
Let be a real Hilbert space with inner product and norm , let be a nonempty closed convex subset of , and let be an operator. Then the variational inequality problem  is to find such that
The literature contains many methods for solving variational inequality problems; see [2–25] and references therein. According to the relationship between the variational inequality problems and a fixed point problem, we can obtain where the projection operator is the projection from onto , that is, In this paper, is an operator with -Lipschtz and -strongly monotone; that is, satisfies the following conditions: If is small enough, then is a contraction. Naturally, the convergence of Picard iterates generated by the right-hand side of (2) is obtained by Banach’s fixed point theorem. Such a method is called the projection method or more results about the projection method see [6, 8, 20] and so forth.
In fact, the projection in the contraction methods may not be easy to compute, and a great effort is to compute the projection in each iteration. Yamada and Deutsch have provided a hybrid steepest-descent method for solving the [2, 3] in order to reduce the difficulty and complexity of computing the projection . Subsequently, the convergence of hybrid steepest-descent methods was given out by Xu and Kim  and Zeng et al. . Naturally, by analyzing several three-step iterative methods in each iteration by the fixed pointed equation, we can obtain the Noor iterations. Recently, Ding et al.  proposed a three-step relaxed hybrid steepest-descent method for variational inequalities, and the simple proof of three-step relaxed hybrid steepest-descent methods under different conditions was introduced by Yao et al. . The literature [14, 16] described a modified and relaxed hybrid steepest-descent (MRHSD) method and the different convergence of the MRHSD method under the different conditions. A set of practical numerical experiments in the literature  demonstrated that the MRHSD method has different efficiency under different conditions. Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method)  makes more use of the history information and less decreases the loss of information than the methods [7, 14]. The PRH method introduced more descent directions than the MRHSD method [14, 16], and computing these descent directions only needs the history information.
In this paper, we will prove the strong convergence of PRH method under different and suitable restrictions imposed on parameters (Condition 12), which differs from that of . Moreover, the proof of strong convergence is different from the previous proof in , which is not similar to that in  in Step 2. And more importantly, numerical experiments verify that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method [14, 16]. Furthermore, it is easy to obtain these descent directions.
The remainder of the paper is organized as follows. In Section 2, we review several lemmas and preliminaries. We prove the convergence theorem under Condition 12 in Section 3. In Section 4, we give out a series of numerical experiments, which demonstrated that the PRH method under Condition 12 is more efficient than under Condition 10. Section 5 concludes the paper.
In order to proof the later convergence theorem, we introduce several lemmas and the main results in the following.
Lemma 1. In a real Hilbert space H, there holds the inequality
The lemma is a basic result of a Hilbert space with the inner product.
Lemma 2 (demiclosedness principle). Assume that is a nonexpansive self-mapping on a nonempty closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .
The following lemma is an immediate result of a projection mapping onto a closed convex subset of a Hilbert space.
Lemma 3. Let be a nonempty closed convex subset of . For and , then(1), (2)−.
Lemma 4 (see ). Let and be bounded sequence in a Banach space X and let be a sequence in with . Suppose for all integers and . Then .
Since is -strongly monotone, has a unique solution . Assume that is a nonexpansive mapping with the fixed point set . Obviously .
For any given numbers and , we define the mapping by
Lemma 6 (see ). If and , then is a contraction. In fact, where , for all .
Lemma 7 (see ). Let be a sequence of nonnegative numbers with and let be sequence of real numbers with . Then
3. Convergence Theorem
Before analyzing the convergence theorem, we first review the PRH method and related results .
Algorithm 8 (see ). Take three fixed numbers , starting with arbitrarily chosen initial points , compute the sequences such that; Prediction Step 1: , Step 2: , Step 3: , Correction Step 4: where is a nonexpansive mapping.
Let and , satisfy the following conditions.
Remark 9. In fact, the PRH method is the MRHSD method when .
Condition 10. One has
We obtain the strong convergence theorem of PRH method for variational inequalities under different assumptions.
Condition 12. One has
Theorem 13. The sequence converges strongly to , and is the unique solution of the . Assume that and , satisfy Condition 12.
Proof. We divide the proof into several steps.
Step 1. , and are bounded. Since is -strongly monotone, (1) has a unique solution , and , , .
A series of computations yields where , where .
Moreover, we also obtain where , subtituting; (14) into (13) and (14) into (12), we immediately obtain Furthermore, It is easy to obtain the following by induction: where , Hence are also bounded.
Step 2. Consider .
Indeed, by a series of computations, we have According to (20) and the prediction step of Algorithm 8, we also obtain Also by the prediction step of Algorithm 8 and (20), (21), we have
so we get Furthermore, Apply , and and (22), (25) to get According to Lemma 4, we obtain Furthermore, by , we also get By (27), (28) and the correction step of Algorithm 8, we immediately conclude that so we get
Step 3. Consider.
Indeed, by the prediction step of Algorithm 8, we have According to the assumption and , then By (32), we immediately obtain
By a series of computations, we can get Hence, by (28), (33), and (34), we also obtain Using Steps 2 and 3, it is easy to obtain the following corollary.
Corollary 14. Consider .
Applying Steps 2 and 3 , one getsso it is easy to see that
Step 4. Consider .
For some , here exits weakly and such that According to , we have By being the unique solution of , we can obtain Since , we immediately conclude that
Step 5. By Step 1 and Lemma 1, we have where and .
Denote We can rewrite (42) as In fact, satisfies Lemma 5; according to we obtain Moreover, by Step 4, we also obtain
Furthermore, by (43), (47), and (48), it is easy to obtain Consequently apply Lemma 5 to obtain
4. Numerical Experiments
The problem considered in this section is where is the matrix Fröbenis norm; that is,
Let , where Let be given symmetric matrices, and asymmetric which differs from previous approaches [9, 21], and it is to be noted that the extended contraction method (EC method)  has much difficulty in computing the examples when is asymmetric, where in element wise:
Then (51) is equivalent to the following variational inequality: So we get
According to Condition 10, we take the following parameter sequences, and let Condition 10 denote the parameter sequences: According to Condition 12, we take the following parameter sequences, and let Condition 12 denote the parameter sequences: Obviously, we have much difficulty in computing the projection of . In order to reduce the difficulty and complexity of computing the projection , we define by where which can be computed without difficulty and the fixed point set of . According to Theorems 11 and 13, the sequences generated by Algorithm 8 under Conditions 10 and 12 are convergent.
The computation begins with ones in MATLAB and stops as soon as . All codes were implemented in MATLAB 7.1 and ran at a Pentium R 1.70G processor, 2G Acer note computer.
We test the problems with , 200, 300, 400, 500, 1000, and 2000. The test results with the PRH method under different conditions are reported in Tables 1, 2, 3, and 4. And the CPU time is in seconds. It is to be noted that the results of extended contraction method are only given out when the iteration step (It) is less than or equal to 100.
Test Examples 1. In this example we generate the data in a similar manner as in . The entries of diagonal elements of are randomly generated in the interval ; the entries of off-diagonal elements of are randomly generated in the interval (Algorithm 1): When and tolerance , the computation time of the proposed method is too long, so the results of the PRH method give out approximate solution with and tolerance in the following. And the extended contraction method (EC method) has much difficulty in computing the examples when is asymmetric. Furthermore, by introducing auxiliary variable, the certain projection method or relaxed-PPA method  can be implemented by these tests.
Test Examples 2. We form the data of the second problems similarly as in the first test examples. The entries of diagonal elements of are randomly generated in the interval ; the entries of off-diagonal elements of are generated from a uniform distribution in the same interval (Algorithm 2):
From Tables 1 to 3, we found that the iteration numbers and CPU time of PRH under Condition 12 are more efficient than that under Condition 10. In Table 4 of our method, the tests’ results give out that the PRH method under some descent directions is more slightly efficient than those of the MRHSD method [14, 16], and it is easy to obtain these descent directions. Furthermore, it is important to find by Tables 2 and 4.
We have proved the strong convergence of PRH method under Condition 12, which differs from Condition 10. The result can be considered as an improvement and refinement of the previous results . And more importantly, numerical experiments demonstrated that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method. How to select parameters of the PRH method for solving variational inequalities is worthy of further investigations in the future.
This research was supported by National Science and Technology Support Program (Grant no. 2011BAH24B06), Joint Fund of National Natural Science Foundation of China and Civil Aviation Administration of China (Grant no. U1233105), and Science Foundation of the Civil Aviation Flight University of China (Grant no. J2010-45).
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