Volume 2013, Issue 1 515902

Research Article

Open Access

The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Haiwen Xu,

Corresponding Author

Haiwen Xu

[email protected]

School of Computer Science, Civil Aviation Flight University of China, Guanghan 618307, China

School of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China nuaa.edu.cn

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Haiwen Xu,

Corresponding Author

Haiwen Xu

[email protected]

School of Computer Science, Civil Aviation Flight University of China, Guanghan 618307, China

School of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China nuaa.edu.cn

Search for more papers by this author

First published: 23 October 2013

https://doi.org/10.1155/2013/515902

Academic Editor: Xu Minghua

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Abstract

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.

1. Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, let K be a nonempty closed convex subset of H, and let F : H → H be an operator. Then the variational inequality problem VI (F, K) [1] is to find x^* ∈ K 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 P_K is the projection from H onto K, that is,

()

In this paper, F : H → H is an operator with F : κ-Lipschtz and η-strongly monotone; that is, F satisfies the following conditions:

()

If β is small enough, then P_K 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 P_K in the contraction methods may not be easy to compute, and a great effort is to compute the projection P_K in each iteration. Yamada and Deutsch have provided a hybrid steepest-descent method for solving the VI (F, K) [2, 3] in order to reduce the difficulty and complexity of computing the projection P_K. Subsequently, the convergence of hybrid steepest-descent methods was given out by Xu and Kim [4] and Zeng et al. [5]. 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. [7] 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. [24]. 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 [16] demonstrated that the MRHSD method has different efficiency under different conditions. Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method) [15] 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 [15]. Moreover, the proof of strong convergence is different from the previous proof in [15], which is not similar to that in [7] 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.

2. Preliminaries

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 T is a nonexpansive self-mapping on a nonempty closed convex subset K of a Hilbert space H. If T has a fixed point, then (I − T) is demiclosed. That is, whenever x_n is a sequence in K weakly converging to some x ∈ K and the sequence (I − T)x_n strongly converges to some y ∈ H, it follows that (I − T)x = y. Here I is the identity operator of H.

The following lemma is an immediate result of a projection mapping onto a closed convex subset of a Hilbert space.

Lemma 3. Let K be a nonempty closed convex subset of H. For all x, y ∈ H and z ∈ K, then

(1)
〈P_K(x) − x, z − P_K(y)〉 ≥ 0,
(2)
−.

Lemma 4 (see [13].)Let {x_n} and {y_n} be bounded sequence in a Banach space X and let {ζ_n} be a sequence in [0,1] with 0 < lim inf _n→∞ζ_n ≤ lim sup _n→∞ζ_n < 1. Suppose x_n+1 = (1 − ζ_n)y_n + ζ_nx_n for all integers n ≥ 0 and lim sup _n→∞(∥y_n+1 − y_n∥ − ∥x_n+1 − x_n∥) ≤ 0. Then limsup _n→∞∥y_n − x_n∥ = 0.

Lemma 5 ([5, 7]). Let {s_n} be a sequence of nonnegative real numbers satisfying the inequality

()

where α_n, τ_n, and γ_n satisfy the following conditions:

(1)
,
(2)
lim _n→∞sup τ_n ≤ 0,
(3)
.

Then lim _n→∞s_n = 0.

Since F is η-strongly monotone, VI (F, K) has a unique solution x^* ∈ K [5]. Assume that T : H → H is a nonexpansive mapping with the fixed point set Fix (T) = K. Obviously Fix (P_K) = K.

For any given numbers λ ∈ (0,1) and μ ∈ (0,2η/κ²), we define the mapping

()

Lemma 6 (see [5].)If 0 < μ < 2η/κ² and 0 < λ < 1, then is a contraction. In fact,

()

where

, for all x, y ∈ H.

Lemma 7 (see [7].)Let {α_n} be a sequence of nonnegative numbers with limsup _n→∞α_n < ∞ and let {β_n} be sequence of real numbers with lim sup _n→∞β_n ≤ 0. Then

()

3. Convergence Theorem

Before analyzing the convergence theorem, we first review the PRH method and related results [15].

Algorithm 8 (see [15].)Take three fixed numbers t, ρ, γ ∈ (0, 2η/κ²), starting with arbitrarily chosen initial points x₀ ∈ H, compute the sequences such that;

Prediction
- Step 1: ,
- Step 2: ,
- Step 3: ,
Correction
- Step 4:
where T : H → H is a nonexpansive mapping.

Let {α_n}⊂[0,1), {β_n}⊂[0,1] and {γ_n}⊂[0,1], satisfy the following conditions.

Remark 9. In fact, the PRH method is the MRHSD method when θ_n ≡ 0, for all n.

Condition 10. One has

()

Theorem 11 (see [15].)In Condition 10, the sequence {x_n} converges strongly to x^* ∈ K, and x^* is the unique solution of the VI(F, K).

We obtain the strong convergence theorem of PRH method for variational inequalities under different assumptions.

Condition 12. One has

()

Theorem 13. The sequence {x_n} converges strongly to x^* ∈ K, and x^* is the unique solution of the VI(F, K). Assume that {α_n}, {β_n} and {γ_n}, satisfy Condition 12.

Proof. We divide the proof into several steps.

Step 1. , and are bounded. Since F is η-strongly monotone, VI (F, K) (1) has a unique solution x^* ∈ K, 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 M₀ = max {3∥x₀ − x^*∥, 3(ρ + γ + t)∥F(x^*)∥/τ},

()

Hence

()

are also bounded.

Step 2. Consider ∥x_n+1 − x_n∥ → 0.

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

()

Let

()

so we get

()

Furthermore,

()

Apply lim _n→∞ β_n = 1, lim _n→∞λ_n = 0, and lim _n→∞γ_n = 1 and (22), (25) to get

()

According to Lemma 4, we obtain

()

Furthermore, by lim _n→∞γ_n = 1, we also get

()

By (27), (28) and the correction step of Algorithm 8, we immediately conclude that

()

so we get

()

Step 3. Consider∥x_n+1 − Tx_n∥ → 0.

Indeed, by the prediction step of Algorithm 8, we have

()

According to the assumption lim _n→∞β_n = 1 and lim _n→∞λ_n = 0, 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 ∥x_n − Tx_n∥ → 0.

Applying Steps 2 and 3 , one gets

()

so it is easy to see that

()

Step 4. Consider .

For some , here exits weakly and such that

()

According to

, we have

()

By x^* being the unique solution of VI (F, K), we can obtain

()

Since

, we immediately conclude that

()

Step 5. By Step 1 and Lemma 1, we have

()

where

()

and M₀ ≪ M < ∞.

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 ∥·∥_F is the matrix Fröbenis norm; that is,

()

Note that the matrix Fröbenis norm is induced by the inner product

()

The problems arise from finance and statistics, and we form the test problems similarly as in [9, 21].

Let

, where

()

Let H_L, H_U be given n × n symmetric matrices, and C asymmetric which differs from previous approaches [9, 21], and it is to be noted that the extended contraction method (EC method) [9] has much difficulty in computing the examples when C is asymmetric, where H_L ≤ H_U 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 P_K[X], for all x ∈ Sⁿ. In order to reduce the difficulty and complexity of computing the projection P_K, we define TX by

()

where

()

which can be computed without difficulty and the fixed point set of Fix (T) = K. According to Theorems 11 and 13, the sequences generated by Algorithm 8 under Conditions 10 and 12 are convergent.

The computation begins with ones (n, n) in MATLAB and stops as soon as ∥x_k+1 − x_k∥ ≤ 10⁻³ or 10⁻². 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 n = 100, 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.

Table 1. Numerical results for the PRH method and the EC method.

Asymmetric matrix	c₀ = 0.1, θ_n = 0.8, tolerance = 10⁻⁴
Asymmetric matrix	Condition 10		Condition 12		EC method
n	It	cpu	It	cpu	It	cpu	tolerance
100	201	8.34	130	5.35	100	14.46	8.289e + 000
200	333	75.44	208	47.14	100	94.30	1.010e + 002
300	443	318.02	272	174.70	100	302.29	4.899e + 002
400	543	789.16	330	446.00	100	686.83	9.628e + 002
500	647	1747.70	388	972.18	100	1287.36	1.756e + 003
1000	1082	19884.30	634	11502.13	100	9220.50	9.826e + 003
2000	>2000	>150000	1052	128504.67	100	>74640.41	>5.597e + 003

Table 2. Numerical results for tolerance 10⁻⁴.

Asymmetric matrix	c₀ = 0.1, θ_n = 0.8
Asymmetric matrix	Condition 10		Condition 12
n	It	cpu	It	cpu
100	204	8.78	130	5.45
200	330	76.08	208	47.72
300	445	323.20	272	175.89
400	548	867.56	330	450.59
500	663	1916.90	388	994.18

Table 3. Numerical results for tolerance 10⁻³.

Asymmetric matrix	c₀ = 0.1, θ_n = 0.8
Asymmetric matrix	Condition 10		Condition 12
n	It	cpu	It	cpu
1000	193	3893.63	126	2280.74
2000	318	42981.02	200	28737.65

Table 4. Numerical results for tolerance 10⁻⁴.

Asymmetric matrix	γ = 0.1, ρ = 0.3, t = 0.1
Asymmetric matrix	θ_n = 0		θ_n = 0.2		θ_n = 0.4		θ_n = 0.6		θ_n = 0.8
n	It	cpu	It	cpu	It	cpu	It	cpu	It	cpu
100	132	5.52	134	5.60	128	5.50	134	5.67	132	5.54
200	210	48.04	206	47.22	208	48.04	204	47.15	214	48.58
300	274	177.49	268	176.08	276	178.80	274	177.68	276	178.84
400	336	468.28	328	445.93	336	468.20	334	454.24	330	453.79
500	392	977.79	394	1012.57	378	948.44	386	953.91	390	971.10

Test Examples 1. In this example we generate the data in a similar manner as in [9]. The entries of diagonal elements of C are randomly generated in the interval (0,2); the entries of off-diagonal elements of C are randomly generated in the interval (−1,1) (Algorithm 1):

()

When n ≥ 1000 and tolerance 10⁻⁴, the computation time of the proposed method is too long, so the results of the PRH method give out approximate solution with n ≥ 1000 and tolerance 10⁻³ in the following. And the extended contraction method (EC method) has much difficulty in computing the examples when C is asymmetric. Furthermore, by introducing auxiliary variable, the certain projection method or relaxed-PPA method [10] can be implemented by these tests.

Algorithm 1

Matlab code:
C = zeros(n, n); HU = ones(n, n)*0.1; HL = −HU;
for i = 1 : n
for j = 1 : n
t = mod (t*42108 + 13846,46273);
C(i, j) = t*2/46273 − 1;
end;
end;
for i = 1 : n
C(i, i) = abs(C(i, i))*2; HU(i, i) = 1; HL(i, i) = 1;
end;

Test Examples 2. We form the data of the second problems similarly as in the first test examples. The entries of diagonal elements of C are randomly generated in the interval (0,2); the entries of off-diagonal elements of C are generated from a uniform distribution in the same interval (Algorithm 2):

()

Algorithm 2

Matlab code:
C = zeros(n, n); HU = ones(n, n)*0.1; HL = −HU;
for i = 1 : n
for j = 1 : n
C = −1 + 2*rand(n);
end;
end;
for i = 1 : n
C(i, i) = abs(C(i, i))*2; HU(i, i) = 1; HL(i, i) = 1;
end;

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 γ, ρ, and t by Tables 2 and 4.

5. Conclusions

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 [14]. 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.

Acknowledgments

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).

References

1 Gowda M. S. and Song Y., On semidefinite linear complementarity problems, Mathematical Programming. (2000) 88, no. 3, 575–587, https://doi.org/10.1007/PL00011387, MR1782157, ZBL0987.90082.
10.1007/PL00011387
Web of Science® Google Scholar
2 Yamada I., D. Bumariu, Y. Censor, and S. Reich, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, 2001, 8, North-Holland, Amsterdam, The Netherlands, 473–504, https://doi.org/10.1016/S1570-579X(01)80028-8, MR1853237, ZBL1013.49005.
10.1016/S1570-579X(01)80028-8
Google Scholar
3 Deutsch F. and Yamada I., Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numerical Functional Analysis and Optimization. (1998) 19, no. 1-2, 33–56, https://doi.org/10.1080/01630569808816813, MR1606921, ZBL0913.47048.
10.1080/01630569808816813
Web of Science® Google Scholar
4 Xu H. K. and Kim T. H., Convergence of hybrid steepest-descent methods for variational inequalities, Journal of Optimization Theory and Applications. (2003) 119, no. 1, 185–201, https://doi.org/10.1023/B:JOTA.0000005048.79379.b6, MR2028445, ZBL1045.49018.
10.1023/B:JOTA.0000005048.79379.b6
Web of Science® Google Scholar
5 Zeng L. C., Wong N. C., and Yao J. C., Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, Journal of Optimization Theory and Applications. (2007) 132, no. 1, 51–69, https://doi.org/10.1007/s10957-006-9068-x, MR2303800, ZBL1137.47059.
10.1007/s10957-006-9068-x
Web of Science® Google Scholar
6 Zeng L. C., On a general projection algorithm for variational inequalities, Journal of Optimization Theory and Applications. (1998) 97, no. 1, 229–235, https://doi.org/10.1023/A:1022687403403, MR1622212, ZBL0907.90265.
10.1023/A:1022687403403
Web of Science® Google Scholar
7 Ding X. P., Lin Y. C., and Yao J. C., Three-step relaxed hybrid steepest-descent methods for variational inequalities, Applied Mathematics and Mechanics. (2007) 28, no. 8, 1029–1036, https://doi.org/10.1007/s10483-007-0805-x, MR2344249, ZBL1231.49004.
10.1007/s10483-007-0805-x
Google Scholar
8 He B. S., A new method for a class of linear variational inequalities, Mathematical Programming. (1994) 66, no. 2, 137–144, https://doi.org/10.1007/BF01581141, MR1297058, ZBL0813.49009.
10.1007/BF01581141
Web of Science® Google Scholar
9 He B. S. and Xu M. H., A general framework of contraction methods for monotone variational inequalities, Pacific Journal of Optimization. (2008) 4, no. 2, 195–212, MR2426625, ZBL1190.90120.
Web of Science® Google Scholar
10 He B. S., PPA-based contraction methods for general linearly constrained convex optimization, Lectures of Contraction Methods for Convex Optimization and Monotone Variational Inequalities, 06C, 2012, http://math.nju.edu.cn/~hebma/.
Google Scholar
11 Huang N. J., Huang X. X., and Yang X. Q., Connections among constrained continuous and combinatorial vector optimization, Optimization. (2011) 60, no. 1-2, 15–27, https://doi.org/10.1080/02331930903353344, MR2772133, ZBL1237.90196.
10.1080/02331930903353344
Web of Science® Google Scholar
12 Harker P. T. and Pang J. S., A damped-Newton method for the linear complementarity problem, Computational Solution of Nonlinear Systems of Equations, 1990, 26, American Mathematical Society, Providence, RI, USA, 265–284, MR1066287.
Google Scholar
13 Suzuki T., Strong convergence of Krasnoselskii and Mann′s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, Journal of Mathematical Analysis and Applications. (2005) 305, no. 1, 227–239, https://doi.org/10.1016/j.jmaa.2004.11.017, MR2128124, ZBL1068.47085.
10.1016/j.jmaa.2004.11.017
Web of Science® Google Scholar
14 Xu H. W., Song E. B., Pan H. P., Shao H., and Sun L. M., The modified and relaxed hybrid steepestdescent methods for variational inequalities, 2, Proceedings of the 1st International Conference on Modelling and Simulation, 2008, World Academic Press, 169–174.
Google Scholar
15 Xu H. W., Shao H., and Zhang Q. C., The Prediction-correction and relaxed hybrid steepest-descent method for variational inequalities, 1, Proceedings of the International Symposium on Education and Computer Science, 2009, IEEE Computer Society and Academy, 252–256.
Google Scholar
16 Xu H. W., Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality, Journal of Inequalities and Applications. (2012) 2012, article 93, https://doi.org/10.1186/1029-242X-2012-93, MR2926331.
10.1186/1029-242X-2012-93
Google Scholar
17 Hammond J. H., Solving asymmetric variational inequality problems and systems of equations with generalized nonlinear programming algorithms [Ph.D. dissertation], 1984, Department of Mathematics, MIT, Cambridge, Mass, USA.
Google Scholar
18 Tseng P., Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Mathematical Programming. (1990) 48, no. 2, 249–263, https://doi.org/10.1007/BF01582258, MR1073710, ZBL0725.90079.
10.1007/BF01582258
Web of Science® Google Scholar
19 Horn R. A. and Johnson C. R., Topics in Matrix Analysis, 1991, Cambridge University Press, Cambridge, UK, https://doi.org/10.1017/CBO9780511840371, MR1091716.
10.1017/CBO9780511840371
Google Scholar
20 Sun D. F., A projection and contraction method for generalized nonlinear complementarity problems, Mathematica Numerica Sinica. (1994) 16, no. 2, 183–194, MR1392612.
Google Scholar
21 Gao Y. and Sun D. F., Calibrating least squares covariance matrix problems with equality and inequality constraints, 2008, Department of Mathematics, National University of Singapore.
Google Scholar
22 Noor M. A., Some recent advances in variational inequalities. I. Basic concepts, New Zealand Journal of Mathematics. (1997) 26, no. 1, 53–80, MR1450808, ZBL0886.49004.
Google Scholar
23 Noor M. A., New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications. (2000) 251, no. 1, 217–229, https://doi.org/10.1006/jmaa.2000.7042, MR1790406, ZBL0964.49007.
10.1006/jmaa.2000.7042
Web of Science® Google Scholar
24 Yao Y., Noor M. A., Chen R., and Liou Y.-C., Strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities, Applied Mathematics and Computation. (2008) 201, no. 1-2, 175–183, https://doi.org/10.1016/j.amc.2007.12.011, MR2432593.
10.1016/j.amc.2007.12.011
Web of Science® Google Scholar
25 A. Nagurney, Advances in Equilibrium Modeling, Analysis, and Computation, Annals of Operations Research, 1993, 44, J. C. Baltzer AG Scientific Publishing, Basel, Switzerland.
Google Scholar

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The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Abstract

1. Introduction

2. Preliminaries

3. Convergence Theorem

4. Numerical Experiments

5. Conclusions

Acknowledgments

References

References

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The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Abstract

1. Introduction

2. Preliminaries

3. Convergence Theorem

4. Numerical Experiments

5. Conclusions

Acknowledgments

References

References

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