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For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.

The complementarity problems have been widely used in the engineering design, information technology, economic equilibrium, etc. Since some elements may involve uncertain data in practical applications, many problems can be attributed to stochastic variational inequality problems or stochastic linear complementarity problems, and arouse the attention of many researchers. Gurkan et al. [

In recent years, some scholars have proposed a series of methods for the study of large-scale complementarity problems. Dong and Jiang [

In this paper, we extend the modulus-based matrix splitting iteration methods to solve the large-scale stochastic linear complementarity problems. We also prove the convergence of these methods when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix. The numerical results show that these methods are efficient.

The outline of the paper is as follows. In Section 2 we present some necessary results and lemmas. In Section 3 we establish the modulus-based matrix splitting iteration methods for solving the SLCP. The convergence of the methods is proved in Section 4. The numerical results are shown in Section 5. Finally, in Section 6, we give some concluding remarks.

In this section, we briefly introduce some necessary results and lemmas.

Let A = ( a i j ) ∈ ℝ n × n , A is said to be positive semi-definite if x T A x ≥ 0 for all x ∈ ℝ n , and positive definite if x T A x > 0 for all x ∈ ℝ n \ { 0 } . A ∈ R n × n is called a P 0 -matrix if all of its principle minors are nonnegative.

Let ( Ω , F , Ρ ) be a probability space, where Ω is a sample subset of ℝ m . Suppose that probability distribution is known, we consider the stochastic linear complementarity problem (SLCP): finding a vector z ∈ ℝ n such that

M ( ω ) z + q ( ω ) ≥ 0 , z ≥ 0 , z T ( M ( ω ) z + q ( ω ) ) = 0 , ω ∈ Ω . (1)

where M ( ω ) ∈ ℝ n × n and q ( ω ) ∈ ℝ n are the rand matrices and vectors for ω ∈ Ω , respectively.

Usually there not exists z for all ω ∈ Ω for Problem (1). In order to get a reasonable solution of (1), in this paper we use the EV formulation proposed by Gurkan et al. [

The Expected Value (EV) Formulation [

Let F ( z , ω ) = M ( ω ) z + q ( ω ) , M ¯ = E [ M ( ω ) ] , q = E [ q ( ω ) ] , and E be the expectation. We consider the following EV formulation: finding a vector z ∈ ℝ n such that

F ¯ ( z ) = E [ F ( z , ω ) ] = M ¯ z + q ≥ 0 , z ≥ 0 , z T F ¯ ( z ) = 0. (2)

We briefly denote it as LCP ( q , M ¯ ) .

Define

RES ( z ) : = min ( z , M ¯ z + q )

where the min operator denotes the componentwise minimum of two vectors. It is generally known that z * solves the LCP ( q , M ¯ ) if and only if z * solves the equations

R E S ( z ) = 0

The function RES is called the natural residual of the LCP ( q , M ¯ ) and is often used in error analysis.

Lemma 1 (see [

( α I + M ¯ ) x = ( α I − M ¯ ) | x | − q (3)

Moreover, if x is the solution of (3), then

r : = α ( | x | − x ) , z : = | x | + x (4)

define a solution pair of Problem (2). On the other hand, if the vector pair z and r solves Problem (2), then x : = 1 / 2 ( z − r / α ) solves the fixed-point problem (3).

In this section, we aim at the EV formulation of the stochastic linear complementarity problem (2). We give some corresponding modulus-based matrix splitting iteration methods.

For the strong monotone stochastic linear complementarity problem, the coefficient matrix is positive definite. For this case, we can apply the method proposed by Dong and Jiang [

Method 3.1

Step 1: Select an arbitrary initial vector x ( 0 ) ∈ ℝ n and set k : = 0 ;

Step 2: Calculate x ( k + 1 ) through the iteration scheme

( α I + M ¯ ) x ( k + 1 ) = ( α I − M ¯ ) | x ( k ) | − q

Step 3: Let z ( k + 1 ) = | x ( k + 1 ) | + x ( k + 1 ) , if z ( k + 1 ) satisfies the termination rule, then stop; otherwise, set k : = k + 1 and return to Step 2.

Unfortunately, the coefficient matrices of some stochastic linear complementarity problems are positive semi-definite, Method 3.1 is not suitable for solving the problem (2). Cottle et al. [

Method 3.2

Step 1: Select a positive number ε 0 ∈ R and an arbitrary initial vector x ε ( 0 ) ∈ ℝ n , and set k : = 0 ;

Step 2: Generate the iteration sequence x ε ( k + 1 ) through solving the following equations

( α I + M ¯ + ε I ) x ε ( k + 1 ) = ( α I − M ¯ − ε I ) | x ε ( k ) | − q .

Let z ε ( k + 1 ) = | x ε ( k + 1 ) | + x ε ( k + 1 ) .

Step 3: Set ε = α ε , where α ∈ ( 0 , 1 ) is a positive number, k : = k + 1 , and return to Step 2.

In this section, we analyze the convergence of Method 3.1 and Method 3.2 when the coefficient matrix of the LCP ( q , M ¯ ) is a symmetric positive definite matrix and a symmetric positive semi-definite matrix.

We first discuss the convergence of Method 3.1 when the coefficient matrix is symmetric positive definite.

Theorem 1 Suppose that the system matrix M ¯ ∈ ℝ n × n is symmetric positive definite, then the sequence { x ( k ) } generated by Method 3.1 converges to x ∗ .

Proof. By Lemma 1 we get

x ( k + 1 ) = ( α I + M ¯ ) − 1 ( α I − M ¯ ) | x ( k ) | − ( α I + M ¯ ) − 1 q .

If x ∗ is a solution of (3), then

x ∗ = ( α I + M ¯ ) − 1 ( α I − M ¯ ) | x ∗ | − ( α I + M ¯ ) − 1 q .

We can get that

‖ x ( k + 1 ) − x ∗ ‖ = ‖ ( α I + M ¯ ) − 1 ( α I − M ¯ ) ( | x ( k ) | − | x ∗ | ) ‖ ≤ ‖ ( α I + M ¯ ) − 1 ( α I − M ¯ ) ‖ ‖ | x ( k ) | − | x ∗ | ‖ ≤ ‖ ( α I + M ¯ ) − 1 ( α I − M ¯ ) ‖ ‖ x ( k ) − x ∗ ‖

Since matrix M ¯ is a symmetric positive definite, we have

‖ ( α I + M ¯ ) − 1 ( α I − M ¯ ) ‖ = max λ i ∈ λ ( M ¯ ) | α − λ i α + λ i | : = σ ( α ) .

where λ ( M ¯ ) denotes the set of all the eigenvalues of M ¯ . As λ i > 0 , it follows that

| α − λ i α + λ i | < 1

and thus

‖ ( α I + M ¯ ) − 1 ( α I − M ¯ ) ‖ = σ ( x ) < 1.

Hence, by the Banach contraction mapping theorem, we have the convergence of the infinite sequence { x ( k ) } to the unique solution x ∗ of the fixed-point equation.

We now discuss the convergence of Method 3.2 when the coefficient matrix is symmetric positive semi-definite.

Lemma 2 (See [

1) If M ¯ ∈ R 0 , then the sequence

2) If

Theorem 2 Suppose that the system matrix

Proof Note that

Then

Let

Moreover,

Therefore

When

By Lemma 2, we have that every accumulation point of

In this section, we test some numerical results to show the efficiency of our methods. Let

Let

n | IT | CPU | RES |
---|---|---|---|

500 1000 2000 3000 3300 | 39^{ } 37 38 43 42 | 0.100 0.825 4.320 10.677 12.448 | 3.60e-06 1.31e-06 3.81e-06 8.57e-06 1.38e-06 |

n | IT | CPU | RES |
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500 1000 2000 3000 3300 | 47^{ } 41 50 44 49 | 0.139 0.933 5.200 11.081 13.711 | 2.88e-06 6.99e-07 4.24e-06 2.53e-06 6.26e-07 |

n | IT | CPU | RES |
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500 1000 2000 3000 3300 | 47^{ } 41 50 44 49 | 0.120 0.800 4.770 10.903 13.730 | 2.88e-06 6.99e-07 4.24e-06 2.53e-06 6.26e-07 |

Method | n = 30 n = 60 n = 150 | ||
---|---|---|---|

CPU | CPU | CPU | |

Method 3.2 FSNM ^{[10]} ^{ } | 0.0078 0.0300 | 0.0156 0.0499 | 0.0188 1.5356 |

In this paper, we study the fast numerical methods for solving the stochastic linear complementarity problems. Firstly, we convert the expected value formulation of stochastic linear complementarity problems into the equivalent fixed point equations, then we establish a class of modulus-based matrix splitting iteration methods, and analyze the convergence of the method. These new methods can be applied to solve the large-scale stochastic linear complementarity problems. The numerical results also show the effectiveness of the new methods.

This work was supported by Natural Science Foundation of China (11661027), National Project for Research and Development of Major Scientific Instruments (61627807), and Guangxi Natural Science Foundation (2015 GXNSFAA 139014).

The authors declare no conflicts of interest regarding the publication of this paper.

Lu, Q.Q. and Li, C.L. (2019) Modulus-Based Matrix Splitting Iteration Methods for a Class of Stochastic Linear Complementarity Problem. American Journal of Operations Research, 9, 245-254. https://doi.org/10.4236/ajor.2019.96016