Journal of Applied Mathematics

Volume 2013 (2013), Article ID 497586, 7 pages

http://dx.doi.org/10.1155/2013/497586

Research Article

Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Mei-Ju Luo¹ and Yuan Lu²

¹School of Mathematics, Liaoning University, Liaoning 110036, China

²School of Sciences, Shenyang University, Liaoning 110044, China

Received 31 January 2013; Revised 10 May 2013; Accepted 14 May 2013

Academic Editor: Farhad Hosseinzadeh Lotfi

Copyright © 2013 Mei-Ju Luo and Yuan Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochastic -function, stochastic -function, and stochastic uniformly -function. Furthermore, the conditions such that the function is a stochastic -function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.

1. Introduction

Given a vector-valued function , the stochastic complementarity problems, denoted by SCP(), are to find a vector such that

where is a random vector with given probability distribution and “a.s." means “almost surely" under the given probability measure. Particularly, when is an affine function of for any , that is,

where and , the is called stochastic linear complementarity problems, denoted by . Correspondingly, problem (1) is called stochastic nonlinear complementarity problem, denoted by if can not be denoted by an affine function of for any . The deterministic problems, which are called complementarity problems (denoted by ), have been intensively studied. More information about theoretical analysis, numerical algorithms and applications especially in economics and engineering can be found in comprehensive books [1, 2].

In practical applications, some elements may involve stochastic factors. In fact, due to stochastic factors, the function value of depends not only on , but also on random vectors. Hence, problem (1) does not have solution in general for almost all . To solve these problems, researchers focus on giving reasonable deterministic reformulations for . Certainly, different deterministic formulations may yield different solutions that are optimal in different senses. In the study of , three types of formulations have been proposed; the expected value (EV) formulation, the expected residual minimization (ERM) formulation, and the SMPEC formulation [3].

The EV formulation is studied by Gürkan et al. [4]. The problem considered in [4] is actually a stochastic variational inequality problem. When applied to the SCP(), the EV model can be stated as follows:

where means expectation with respect to .

The ERM model is first proposed by Chen and Fukushima [5] for solving the . By employing an NCP function , the (1) is transformed equivalently to the stochastic equations

where is defined by

and denotes the th component of the vector . Here is an NCP function which has the property

Then the ERM formulation for (1) is given by

The NCP functions employed in [5] include the Fischer-Burmeister function, which is defined by

and the min function

In particular, it is known [6, 7] that there exist the following relations between these two functions:

As observed in [5], the ERM formulations with different NCP functions may have different properties. Subsequently, the ERM formulation for has been studied in [6, 8–13]. Note that Fang et al. [8] propose a new concept of stochastic matrice: is called a stochastic matrix if

Moreover, Zhang and Chen [11] introduce a new concept of stochastic function, which can be regarded as a generalization of the stochastic matrix given in [8].

Throughout this paper, we suppose that the sample space is nonempty and compact set and that the function is continuous with respect to and . On the other hand, we will use the following notations: and for a given vector . represents the set for natural numbers and with . for any given vector . refers to the Euclidean norm.

The remainder of the paper is organized as follows: in Section 2, we introduce the concepts of a stochastic -function, a stochastic -function, and a stochastic uniformly -function, which can be regarded as a generalization of the deterministic , -function, and uniformly -function or an extension of stochastic matrix and stochastic matrix [14]. In addition, some properties of a stochastic -function are given. In Section 3, we show the sufficient conditions for the solution set of ERM problem to be nonempty and bounded. In Section 4, we discuss error bounds of . In Section 5, an algorithm will be given to solve ERM model. We then give conclusions in Section 6.

2. Stochastic -Function

It is well known that the -function, -function, and uniformly -function play an important role in the nonlinear complementarity problems theory [1]. We will introduce a new concept of stochastic -function, -function, and uniformly -function, which can be regarded as a generalization of their deterministic form or stochastic matrix and stochastic matrix.

Definition 1 (see [14]). is called a stochastic -matrix if there exists such that, for every in ,

Definition 2. A function is a stochastic ()-function if there exist , such that, for every in ,

Definition 3. A function is a stochastic uniformly -function if there exists a positive constant and , such that, for every in ,

Clearly, every stochastic uniformly -function must be a stochastic -function, which in turn must be a stochastic -function. We further cite the definition of “equicoercive" in [11]. More information about this definition can be found in [11].

Definition 4 (see [11]). We say that is equicoercive on , if, for any satisfying , the existence of with for some implies that

where

and and is the distribution function of .

More details about were included in [8].

Proposition 5. If is a stochastic -function, then is a stochastic -function for every .

Proof. From the definition of stochastic -function, there exist , such that, for every ,

Hence, we have

This proposition gives the relationship between stochastic -function and stochastic -function.

Proposition 6. Let be an affine function of for any defined by (2). Then is a stochastic ()-function if and only if is a stochastic matrix.

Proof. By the definition of stochastic -function, we have that there exist , such that, for every ,

which is equivalent to

when is defined by (2). Set ; then , and we have that formulation (20) holds if and only if there exists such that, for every ,

Hence, is a stochastic matrix.

Proposition 7. is a stochastic -function if and only if there exists a such that is a ()-function.

Proof. For the “if” part, suppose on the contrary that is not a stochastic -function, and then there exist in for any satisfying

On the other hand, since is a -function, then for there exist , such that

Notice that , by the definition of in (16), we have

This contradicts formulation (22). Therefore, is a stochastic ()-function.

Now for the “only if” part, suppose on the contrary that there does not exist a such that is a -function. Then for any , , there exists in , such that

which means that

By the definition of in (16), we have . Hence, formulation (26) is equivalent to

which contradicts definition (20). Therefore, there exists a such that is a ()-function.

Theorem 8. Suppose that is a ()-function. Then is a stochastic -function.

Proof. Suppose on the contrary that is not a stochastic ()-function, then there exist in for any satisfying

This means that

always holds for any and . Furthermore, following from (29), we have

that is

which contradicts the definition of ()-function. Therefore, is a stochastic ()-function.

Note that there is at most one solution (may not be a solution) for the EV model stochastic complementarity problems if is a ()-function.

3. Boundedness of Solution Set

Theorem 9. Suppose that is a stochastic uniformly -function and is equicoercive on . Then the solution set of ERM model (7) defined by and is nonempty and bounded.

Proof. Suppose on the contrary that the ERM model defined by is not bounded. Thus there exist a sequence with and a constant , such that

Define the index set by

By assumption, we have . We now define a sequence as follows:

From the definition of and the fact that is a stochastic uniformly -function, we obtain that for any , there exists such that

and hence there are satisfying

Take subsequence such that the corresponding subscript of (36) is . Noting that and taking (36) into account, we have

from which we get

By definition, the sequence remains bounded. From the continuity of , it follows that the sequence is also bounded for every . Hence, taking a limit in (38), we obtain that there is at least one index such that

Since is equicoercive on , we have

Let

Then . By Fatou’s Lemma [15], we have

Since

on and , then the left-hand side of formulation (42) is infinite. Therefore,

Moreover, it is easy to find

as . This contradicts formulation (32). Hence, the solution set of ERM model (7) defined by is nonempty and bounded. Similar results about can be obtained by relation formulation (10).

4. Robust Solution

As we show, both EV model and ERM model give decisions by a deterministic formulation. However, the decisions may not be the best or may be even infeasible for each individual event. In fact, we should take risk into account to make a priori decision in many cases. Naturally, it is necessary to know how good or how bad the decision which we have given can be. In this section, we study the robustness of solutions of the ERM model. Let denote the solution set of , and define the distance from a point to the set by

Theorem 10. Assume that , and takes values with respective probabilities . Furthermore, suppose that for every , is uniformly P-function and Lipschitz continuous with respect to . Then there is a constant such that

where is defined by or .

Proof. For any fixed , since is uniformly -function and Lipschitz continuous, from Corollary 3.19 of [16], we have unique solution of CP(), and there exists a constant such that

Letting , we have

where the first inequality follows from Cauchy-Schwarz inequality, the second inequality follows from formulation (48), and the third inequality follows from formulation (10). This completes the proof of the theorem.

Theorem 10 particularly shows that for the solution of (7),

This inequality indicates that the expected distance to the solution set for is also likely to be small at the solution of (7). In other words, we may expect that a solution of the ERM formulation (7) has a minimum sensitivity with respect to random parameter variations in SCP(). In this sense, solutions of (7) can be regarded as robust solutions for SCP().

5. Quasi-Monte Carlo and Derivative-Free Methods for Solving ERM Model

Note that the ERM model (7) included an expectation function, which is generally difficult to be evaluated exactly. Hence in this section, we first employ a quasi-Monte Carlo method to obtain approximation problems of (7) for numerical integration. Then, we consider derivative-free methods to solve these approximation problems.

By the quasi-Monte Carlo method, we obtain the following approximation problem of (7):

where is a set of observations generated by a quasi-Monte Carlo method such that and stands for the probability density function. In the rest of this paper, we assume that the probability density function is continuous on . For each , is continuously differentiable function. We denote by the optimal solutions of approximation problems (51). We are interested in the situation where the first-order derivatives of cannot be explicitly calculated or approximated.

Condition 1. Given a point , the level set

is compact.

Condition 2. If and are sequences of points such that , converging to some and for all , then

where and .

Condition 3. For every there exist scalars , and such that

Condition 4. Given and , the set of search directions

satisfing is uniformly bounded and . Here,

Under Conditions 1, 2, and 3 and by choosing satisfying Condition 4 with , then the following generated iterates have at least one cluster point that is a stationary point of (51) for each .

Algorithm 11. Parameters: , , , , , .

Step   1. Set .

Step   2. Choose a set of directions satisfying Condition 4.

Step   3.

(a) Set , , . (b) Compute the maximum stepsize such that for all . Set . (c) If and , set ; otherwise set . (d) If , set , and go to Step 4. (e) If , set , and go to Step 3(b). Otherwise set and go to Step 4.

Step   4. Find such that . Set , and go to Step 2.

For this algorithm, it is easy to proof that if is the sequence produced by algorithm under Conditions 1–4, then is bounded and there exists at least one cluster point which is a stationary point of problem (51) for each .

6. Conclusions

The SCP() has a wide range of applications in engineering and economics. Therefore, it is meaningful and interesting to study this problem. In this paper, we give the definitions of stochastic -function, stochastic -function and stochastic uniformly -function, which can be regarded as a generalization of the deterministic formulation or an extension of a stochastic function given in [11]. Moreover, we consider the conditions when the function is a stochastic ()-function. Furthermore, we show that the involved function being a stochastic uniformly -function and equi-coercive [11] are sufficient conditions for the solution set of the expected residual minimization problem to be nonempty and bounded. Finally, we illustrate that the ERM formulation produces robust solutions with minimum sensitivity in violation of feasibility with respect to random parameter variations in SCP(). On the other hand, we employ a quasi-Monte Carlo method to obtain approximation problems of (7) for dealing numerical integration and further consider derivative-free methods to solve these approximation problems.

Acknowledgments

This work was supported by NSFC Grants no. 11226238 and no. 11226230, and predeclaration fund of state project of Liaoning university 2012, 2012LDGY01, and University Scientific Research Projects of School of Education Department of Liaoning Province 2012, 2012427.

References

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003.

R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1992.

G.-H. Lin and M. Fukushima, “New reformulations for stochastic nonlinear complementarity problems,” Optimization Methods & Software, vol. 21, no. 4, pp. 551–564, 2006.

G. Gürkan, A. Y. Özge, and S. M. Robinson, “Sample-path solution of stochastic variational inequalities,” Mathematical Programming, vol. 84, no. 2, pp. 313–333, 1999.

X. Chen and M. Fukushima, “Expected residual minimization method for stochastic linear complementarity problems,” Mathematics of Operations Research, vol. 30, no. 4, pp. 1022–1038, 2005.

X. Chen, C. Zhang, and M. Fukushima, “Robust solution of monotone stochastic linear complementarity problems,” Mathematical Programming, vol. 117, no. 1-2, pp. 51–80, 2009.

P. Tseng, “Growth behavior of a class of merit functions for the nonlinear complementarity problem,” Journal of Optimization Theory and Applications, vol. 89, no. 1, pp. 17–37, 1996.

H. Fang, X. Chen, and M. Fukushima, “Stochastic matrix linear complementarity problems,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 482–506, 2007.

G.-H. Lin, X. Chen, and M. Fukushima, “New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC,” Optimization, vol. 56, no. 5-6, pp. 641–953, 2007.

C. Ling, L. Qi, G. Zhou, and L. Caccetta, “The property of an expected residual function arising from stochastic complementarity problems,” Operations Research Letters, vol. 36, no. 4, pp. 456–460, 2008.

C. Zhang and X. Chen, “Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty,” Journal of Optimization Theory and Applications, vol. 137, no. 2, pp. 277–295, 2008.

C. Zhang and X. Chen, “Smoothing projected gradient method and its application to stochastic linear complementarity problems,” SIAM Journal on Optimization, vol. 20, no. 2, pp. 627–649, 2009.

G. L. Zhou and L. Caccetta, “Feasible semismooth Newton method for a class of stochastic linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 379–392, 2008.

X. L. Li, H. W. Liu, and Y. K. Huang, “Stochastic matrix and matrix linear complementarity problem,” Journal of Systems Science and Mathematical Sciences, vol. 31, no. 1, pp. 123–128, 2011.

K. L. Chung, A Course in Probability Theory, Academic Press, New York, NY, USA, 2nd edition, 1974.

B. Chen and P. T. Harker, “Smooth approximations to nonlinear complementarity problems,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 403–420, 1997.