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In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.

Suppose

where

If

where

In this paper, we consider the following generalized stochastic linear complementarity problems. Denote

(2)

Let

In the following of this paper, we consider to give a new conjugate gradient projection method for solving (2). The method is based on a suitable reformulation. Base on the Fischer-Burmeister function, x is a solution of (3)

Define

Then solving (3) is equivalent to find a global solution of the minimization problem

So, (3) and (4) can be rewritten as

where

Let

Let

If (2) has a solution, then solving (5) is equivalent to find a global solution of the following minimization problem

where

In this section, we give some Lemmas, which are taken from [

Lemma 1. Let P be the projection onto Ω, let

1)

2) P is a non-expansive operator, that is,

3)

Lemma 2. Let

1)

2) The mapping

3) The point

In this section, we give a new conjugate gradient projection method and give some discussions about this method.

Given an iterate

where

with

And

Method 1. Conjugate Gradient Projection Method (CGPM)

Step 0: Let

Step 1: Compute

Set

Step 2: If

Step 3: Let

In order to prove the global convergence of the Method 1, we give the following assumptions.

Assumptions 1

1) _{1} is initial point.

2) _{0}, and its gradient is Lipschitz continuous, that is, there exists a positive constant L such that

Lemma 3. If t_{k} is not the stability point of (6), _{k} generated by (9) descent

direction, which is

Proof. From (7), Lemma 1, and (8), we have

Lemma 4. [

Theorem 1. Let

nuous on the Ω,

point of

Proof. By Lemma 2, we have

for

Let

By the above formula, (8) and Lemma 1, we get

Taking limit on both sides and by Lemma 4, we know that

Because

and Lemma 4, we have

By (12), (13), (14) and

By (10), we know that

Let

From Lemma 2 3), we get any accumulation point of

In this section, we give the numerical results of the conjugate gradient projection method for the following given test problems, which are all given for the first time. We present different initial point t_{0}, which indicates that Method 1 is global convergence.

Throughout the computational experiments, according to Method 1 for determining the parameters, we set the parameters as

The stopping criterion for the method is

In the table of the test results, t_{0} denotes initial point,

Example 1. Considering SGLCP with

The test results are listed in “

Problem | t_{0} | val | Itr | |
---|---|---|---|---|

Example 1 | 3.3 × 10^{−3} | 1465 | ||

3.3 × 10^{−3} | 1701 | |||

3.3 × 10^{−3} | 2670 | |||

3.3 × 10^{−3} | 3261 | |||

3.3 × 10^{−3} | 3847 | |||

3.3 × 10^{−3} | 4704 | |||

Example 2 | 0.7299 | 62788 | ||

0.7299 | 65528 | |||

0.7299 | 66962 | |||

0.7299 | 100,000 | |||

0.7299 | 100,000 | |||

0.7299 | 100,000 |

Example 2. Considering SGLCP with

The test results are listed in “

In this paper, we present a new conjugate gradient projection method for solving stochastic generalized linear complementarity problems. The global convergence of the method is analyzed and numerical results show that Method 1 is effective. In future work, large-scale stochastic generalized linear complementarity problems need to be studied and developed.

This work is supported by National Natural Science Foundation of China (No. 11101231, 11401331), Natural Science Foundation of Shandong (No. ZR2015AQ013) and Key Issues of Statistical Research of Shandong Province (KT15173).

Zhimin Liu,Shouqiang Du,Ruiying Wang, (2016) A New Conjugate Gradient Projection Method for Solving Stochastic Generalized Linear Complementarity Problems. Journal of Applied Mathematics and Physics,04,1024-1031. doi: 10.4236/jamp.2016.46107