We establish the strong convergence of predictioncorrection and relaxed hybrid steepestdescent 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 steepestdescent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.
Let
The literature contains many methods for solving variational inequality problems; see [
In fact, the projection
In this paper, we will prove the strong convergence of PRH method under different and suitable restrictions imposed on parameters (Condition
The remainder of the paper is organized as follows. In Section
In order to proof the later convergence theorem, we introduce several lemmas and the main results in the following.
In a real Hilbert space H, there holds the inequality
The lemma is a basic result of a Hilbert space with the inner product.
Assume that
The following lemma is an immediate result of a projection mapping onto a closed convex subset of a Hilbert space.
Let
Let
Let
Since
For any given numbers
If
Let
Before analyzing the convergence theorem, we first review the PRH method and related results [
Take three fixed numbers
Prediction
Step 1:
Step 2:
Step 3:
Correction
Step 4:
where
Let
In fact, the PRH method is the MRHSD method when
One has
In Condition
We obtain the strong convergence theorem of PRH method for variational inequalities under different assumptions.
One has
The sequence
We divide the proof into several steps.
A series of computations yields
Moreover, we also obtain
Indeed, by a series of computations, we have
Let
so we get
Indeed, by the prediction step of Algorithm
By a series of computations, we can get
For some
Denote
Furthermore, by (
The problem considered in this section is
Note that the matrix Fröbenis norm is induced by the inner product
Let
Then (
According to Condition
The computation begins with ones
We test the problems with
Numerical results for the PRH method and the EC method.
Asymmetric matrix 



Condition 
Condition 
EC method  

It  cpu  It  cpu  It  cpu  tolerance 
100  201  8.34  130  5.35  100  14.46 

200  333  75.44  208  47.14  100  94.30 

300  443  318.02  272  174.70  100  302.29 

400  543  789.16  330  446.00  100  686.83 

500  647  1747.70  388  972.18  100  1287.36 

1000  1082  19884.30  634  11502.13  100  9220.50 

2000 


1052  128504.67  100  >74640.41  >5.597 
Numerical results for tolerance
Asymmetric matrix 



Condition 
Condition 


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 
Numerical results for tolerance
Asymmetric matrix 



Condition 
Condition 


It  cpu  It  cpu 
1000  193  3893.63  126  2280.74 
2000  318  42981.02  200  28737.65 
Numerical results for tolerance
Asymmetric matrix 










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 
Matlab code:
for
for
end;
end;
for
end;
Matlab code:
for
for
end;
end;
for
end;
From Tables
We have proved the strong convergence of PRH method under Condition
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. J201045).