Abstract
This paper uses a merit function derived from the Fishcher–Burmeister function and formulates box-constrained stochastic variational inequality problems as an optimization problem that minimizes this merit function. A sufficient condition for the existence of a solution to the optimization problem is suggested. Finally, this paper proposes a Monte Carlo sampling method for solving the problem. Under some moderate conditions, comprehensive convergence analysis is included as well.
1. Introduction
Let and be two -dimensional vectors with components and satisfying , and denote, by , the nonempty and possibly infinite box . Then, the box-constrained variational inequality problem (BVIP, for short) is to find a vector such thatwhere is a given function. This problem is also called the mixed complementarity problem [1]. Both linear complementarity problems [2] and nonlinear complementarity problems [3] have played an important role in studying economic equilibrium and engineering problems [4, 5].
Much effort has been made to derive merit functions for the BVIP and then, using these functions, to develop solution methods. Recently, Kebaili and Benterki [6] propose a penalty approach for a box-constrained variational inequality problem (BVIP). It is replaced by a sequence of nonlinear equations containing a penalty term. A homotopy method for solving mathematical programs with box-constrained variational inequalities is presented [7]. A reformulation of the BVIP, based on the Fishcher–Burmeister function, is described in paper [8] by Sun and Womersley. Further reformulations can be obtained by replacing the Fishcher–Burmeister function by the introduced function from [9] which seems to have somewhat stronger theoretical properties and a better numerical behavior.
Stochastic variational inequality problem (SVIP, for short) model is a natural extension of deterministic variational inequality models. Over the past few decades, deterministic variational inequality has been extensively studied for its extensive application in engineering, economics, game theory, and networks; see the book on the topic by Facchinei and Pang [1]. While many practical problems only involve deterministic data, there are some important instances where problem data contain some uncertain factors and consequently SVIP models are needed to reflect uncertainties. Gürkan et al. [10] have shown how to extend a simulation-based method, sample-path optimization, to solve SVIP, and Robinson [11] has provided a mathematical justification for sample-path optimization, while Shapiro et al. [12] suggested Monte Carlo sampling methods for SVIP. SVIP can be found in pricing game [13] and inventory competition [14] among several firms that provide substitutable goods or services. Some stochastic dynamic noncooperative games [15] and competitive Markov decision processes [16] can be formulated as examples of SVIP. Jiang and Xu [17] proposed a stochastic approximation method for numerical solution of SVIP. The method is an iterative scheme where, at each iterate, a correction is made and the correction is obtained by sampling or other stochastic approximation. Xu [18] applied the well-known sample average approximation (SAA, for short) method to solve the same class of stochastic variational inequality problems (SVIP).
Aimed at a practical treatment of the SVIP, box-constrained stochastic variational inequality problem (BSVIP, for short) is meaningful and interesting to study [19]. Motivated by Sun and Womersley [8], Luo and Lin [20] formulated a class of BSVIP as an optimization problem that minimizes the expected residual of the merit function derived on the Fishcher–Burmeister function and proposed a Monte Carlo sampling method for solving the problem.
In this paper, we consider the following BSVIP to find a vector such thatwhere is a random vector defined on probability space , is a mapping, denotes the mathematical expectation with respect to the distribution of , and “a.s.” is the abbreviation for “almost surely” under the given probability measure. To ease the notation, we will write as , and this should be distinguished from being a deterministic vectors of in the context.
We are concerned with the numerical solution of BSVIP. If we are able to obtain a closed form of , then BSVIP becomes a deterministic BVIP, and the existing numerical methods for the latter [8] can be applied directly. However, in practice, obtaining a closed form of or computing the value of it numerically is often difficult either due to the unavailability of the distribution of or because it involves multiple integration.
Motivated by the above work, we make use of a merit function derived from the Fishcher–Burmeister function (FB-function, for short) and formulate BSVIP as an optimization problem that minimizes this merit function. We study a sufficient condition for the existence of a solution to the optimization problem. Finally, we propose a Monte Carlo sampling method for solving the problem. Under some moderate conditions, comprehensive convergence analysis is included as well.
The organization of this paper is as follows. In Section 2, we study some preliminary knowledge. Section 3 shows under what conditions the level sets of the merit function are bounded. In Section 4, we make use of a Monte Carlo sampling method to handle the expectation; moreover, we establish convergence of global optimal solutions of approximation problems generated by the proposed method. Preliminary numerical results are reported in Section 5. Finally, we give some conclusions.
2. Preliminaries
We first consider a merit function derived from the NCP-function [21] for BVIP (1). It is easy to see that BVIP is equivalent to its Karush–Kuhn–Tucker (KKT, for short) system:where is a -dimensional unit vector. If solves BVIP, then solve the KKT system (3). Conversely, if solve the KKT system (3), then solves BVIP (1).
Let be a NCP-function. Then, the KKT system (3) can be written as(i)If , then .(ii)If , then , and we have(iii)If , then , and we have(iv)If , thenand we have
Based on , we let as
Then, it is easy to see that
In turn, corresponding to this reformulation of the BVIP, we may define the merit function:
Therefore, is an unconstrained differentiable merit function for BVIP (1). In order to define this merit function, we may consider the FB-function, which is given byand let us introduce a partition of the index set I:
Consequently, we give the merit function for BSVIP (2) as follows:
That is to say, we formulate BSVIP as the following optimization problem:
Throughout, we assume that is continuous with respect to for any . Furthermore, suppose that the sample space is nonempty, and for every ,where means the Euclidean norm. By the above assumption and Th16.8 in [22], we get that is continuous with respect to .
3. Boundedness of Level Sets
In this section, we discuss conditions for boundedness of the level sets of the merit function (13). Consider the level set defined bywhere is a given scalar.
Definition 1. A function is a uniform -function if there exists a positive constant such that, for every x and y in ,Now let us focus on the properties of FB-function .
Lemma 1. For given , we have FB-function satisfying
Proof. From Tseng [23], for any two numbers , we haveThen,Then, we study the boundedness of the level sets.
Theorem 1. Suppose that is a uniform -function. Then, for any , is bounded.
Proof. Suppose that there is a nonnegative number such that is unbounded. This implies that there exists a sequence such that . We first define the index set byBy assumption, we have . We now define a new sequence as follows:From the definition of and the fact that is a uniform -function, we obtain that, for some constant ,from which we obtainBy definition, the sequence remains bounded. From the continuity of , it follows that the sequence is also bounded for every . Hence, we deduce from (24) that there is at least one index such thatIn what follows, we will show thatNow, we consider above three cases, respectively. Case 1(): obviously, . Case 2(): by Lemma 1, we have Case 3: if , it follows from (25) that Case 4: if , then, by (25), we have Case 5(): by Lemma 1, we have Case 6: if , it follows from (25) that Case 7: if , then, by (25), we haveBased on the above fact, we haveThis is a contradiction, and hence, is bounded for any .
Since is nonempty, we see from Theorem 1 that problem (14) has at least one optimal solution when is a uniform -function on .
Note that the expectation function of problem (14) is generally difficult to evaluate exactly. In what follows, we employ a Monte Carlo sampling method for numerical integration to address this question.
4. Monte Carlo Sampling Method and Convergence Analysis
We can view the generated sample , as a sequence of random vectors, each having the same probability distribution as . If the generated random vectors are stochastically independent of each other, we say that the sample is independent identically distributed (iid). With the generated sample , we associate the sample average function:where , and consider the following approximation problem of (14):
We refer (14) as the true problem and (35) as the Monte Carlo sampling average approximation problem.
If the sample is iid, then the law of large numbers (LLN) holds pointwise. We say that the LLN holds, for , pointwise if converges w.p.1 to , as , for any fixed . See [12, 24, 25], for more details about the Monte Carlo sampling method.
Theorem 2. Suppose is continuous with respect to for any and ; if solves problem (35) for each and is an accumulation point of as N tends to infinity, then is an optimal solution of the true problem (14) with probability one.
Proof. Let be a compact set which w.p.1 contains a neighborhood of . We first show that converges w.p.1 to for any . Recall (13) and (34), and we haveNote thatSince is a compact set and is continuous with respect to for any , hence, for every , there exists a constant such thatCombining (37) and (38), we haveSimilarly, we can have that, for every , there exists constant and such thatLet ; by (39), (40), and (41), then converges w.p.1 to , as , for any fixed ; then, for any , we have that w.p.1 for is large enough:Obviously, for every , we can haveFor , we have from (44) thatSince as and is continuous, we haveCombining (47) and (45), then there exists such that w.p.1 for :Note thatCombining (46) and (48), we have that w.p.1 for sufficiently large :This implies that w.p.1 a global minimizer of (35) becomes a global minimizer of (14); hence, it is concluded.
Section 5 will demonstrate the proposed approach.
5. Numerical Results
In this section, we used the notation and example in Wang et al. [26] to illustrate the model of BSVIP and the formulation. In our experiments, we used the command random in Matlab R2010a to generate pseudorandom sequences and employed fmincon to solve problem (35).
Example 1. (see [26]). Consider the stochastic variational inequality problem (2), in which is uniformly distributed on , and is given byNamely, . This problem has a solution for each . The numerical results are shown in Table 1.
Example 2. (see [26]). Consider the stochastic variational inequality problem (2), in which is uniformly distributed on and is given byNamely, . It is easy to prove that the function is strongly monotonous. So, this stochastic variational inequality problem has a unique solution for each . The numerical results are shown in Table 2.
From the above analysis for Examples 1-2, our preliminary numerical results for these examples indicate that the proposed method yields a reasonable and better solution of the stochastic variational inequality problem (2).
6. Conclusions
In this paper, the well-known Mote Carlo sampling method is applied to solve a class of box-constrained stochastic variational inequality problems. Firstly, a merit function derived from the FB-function is used and BSVIP is formulated as an optimization problem that minimizes this merit function. A sufficient condition is suggested for the existence of a solution to the optimization problem. Finally, this paper proposes a Monte Carlo sampling method for solving the problem. Under some moderate conditions, comprehensive convergence analysis is included as well.
Data Availability
The data used to support the findings of this study are from reference [26].
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (Grant no. 11501476) and the Southwest Minzu University Research Startup Funds.