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The stochastic generalized equation provides a unifying methodology to study several important stochastic programming problems in engineering and economics. Under some metric regularity conditions, the quantitative stability analysis of solutions of a stochastic generalized equation with the variation of the probability measure is investigated via Brouwer’s fixed point theorem. In particular, the error bounds described by Hausdorff distance between the solution sets are established against the variation of the probability measure. The stability results obtained are finally applied to a stochastic conic programming.

In this paper, we focus on the following stochastic generalized equation (SGE): find

Model (

In this paper, we concentrate our research on the stability of (

Shapiro et al. [

In this paper, we follow Liu et al.’s approach to investigate the existence and quantitative stability analysis of solutions of problem (

This paper is organized as follows: Section

Throughout this paper we use the following notations. Let

Given a closed set

If

For set-valued maps, the definition of coderivative was introduced by Mordukhovich in [

Consider a mapping

One of the tasks of variational analysis is to detect the stability of a nonlinear system when perturbations of the data occur. For this purpose, the following notations are related; see [

Consider the multifunction

(Metric regularity) we say that

(Lipschitz-like property) we say that

(Calmness) we say that

We know from the definition that the calmness property is weaker than the Lipschitz-like property. As shown in [

Let

Let

In this section, we discuss existence and quantitative stability analysis of solutions to the perturbed SGE (

Let

We know from [

We now turn to discuss existence of a solution to the perturbed SGE (

Suppose that

We know from metric regularity of

In the case when

Suppose that

We only need to verify the conditions in Theorem

We make some comments on the conditions in Corollary

In this section, we undertake stability analysis of SGE (

Suppose that

For any small positive number

for any

If

for any

If

for any

(i) Notice that

Part (ii): since

(iii) We only need to show that there exists

In [

In the case when SGE (

Suppose that condition (b) in Assumption

In [

Suppose that condition (b) in Assumption

there exists

there exist constants

for all

Let

Consider the following stochastic optimization problem:

In this section, we focus on a special case when the probability measure

If condition (b) in Assumption

Let

If condition (b) in Assumption

We at first show that there exists a number

Let

If condition (b) in Assumption

There exists

for

If, in addition, the matrix

is semidefinite for any

for

It suffices to verify the conditions in Theorem

We know from [

The existence results and quantitative stability analysis described by Hausdorff distance are established in this paper for SGE (

The authors declare that they have no conflicts of interest.

This paper is supported by the NSFC under Projects no. 11671183 and no. 11671184 and Program for Liaoning Innovation Talents in University under Project no. LR2017049.