We consider the expected residual minimization method for a class of stochastic quasivariational inequality problems (SQVIP). The regularized gap function for quasivariational inequality problem (QVIP) is in general not differentiable. We first show that the regularized gap function is differentiable and convex for a class of QVIPs under some suitable conditions. Then, we reformulate SQVIP as a deterministic minimization problem that minimizes the expected residual of the regularized gap function and solve it by sample average approximation (SAA) method. Finally, we investigate the limiting behavior of the optimal solutions and stationary points.

The quasivariational inequality problem is a very important and powerful tool for the study of generalized equilibrium problems. It has been used to study and formulate generalized Nash equilibrium problem in which a strategy set of each player depends on the other players’ strategies (see, for more details, [

QVIP is to find a vector

In most important practical applications, the function

Due to the introduction of randomness, SQVIP (

Because of the existence of a random element

Recently, one of the mainstreaming research methods on the stochastic variational inequality problem is expected residual minimization method (see [

In this paper, we focus on ERM method for SQVIP. We first show that the regularized gap function for QVIP is differentiable and convex under some suitable conditions. Then, we formulate SQVIP (

The rest of this paper is organized as follows. In Section

Throughout this paper, we use the following notations.

The regularized gap function for the QVIP (

Let

Though the regularized gap function

The regularized gap function (or residual function) for SQVIP (

Note that the objective function

If

Let

In the remainder of this paper, we restrict ourself to a special case, where

Let

For any

It is easy to know that problem (

If

Let us define the function

When

Now we investigate the conditions under which

Suppose that

If

If

Substituting

If

If

When

When

In this section, we consider the properties of the objective function

Suppose that

Since

In a similar way to Theorem

The following theorem gives some conditions under which

Suppose that the assumption of Theorem

If

If

Define

If

If

It is easy to verify that

In this section, we will investigate the limiting behavior of the optimal solutions and stationary points of (

Note that if the conditions of Theorem

Suppose that the conditions of Theorem

Without loss of generality, we assume that

We first show that

Now, we show that

In general, it is difficult to obtain a global optimal solution of problem (

Let

Without loss of generality, we assume that

At first, we show that

Next, we show that

Now we show that

This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (111171237, 71101099).