Stability Analysis of Stochastic Generalized Equation via Brouwer ’ s Fixed Point Theorem

The stochastic generalized equation provides a unifyingmethodology 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.


Introduction
In this paper, we focus on the following stochastic generalized equation (SGE): find  ∈ R  such that 0 ∈ E  [ (,  ())] + G () , where  : R  × Ξ → R  is a continuous function,  : Ω → Ξ is a random vector defined on a probability space (Ω, F, ) with support set Ξ ⊆ R  , E  [⋅] denotes the expected value with respect to , and G : R  Î R  is an outer semicontinuous set-valued mapping.Throughout the paper, we assume that E  [(, ())] is well defined for any  ∈ R  .
To ease notation, we will use  to denote either the random vector () or an element of R  depending on the context.Model (1) a natural extension of deterministic parametric generalized equation [1] and the study of stochastic generalized equations can be traced down to King and Rockafellar's early work [2].In a particular case when G(⋅) is a normal cone operator N K (⋅) in which K is a closed convex cone in R  , (1) reduces to a stochastic variational inequality problem (SVIP) which has been intensively studied over the past few years; see for instance [3][4][5][6][7] and the references therein.The research ranges from numerical schemes such as stochastic approximation method and Monte Carlo method to the fundamental theory and applications.
In this paper, we concentrate our research on the stability of (1); namely, we look into the impact of variation of probability measure  on the solution of the SGE.Like similar existing research in deterministic generalized equation, this kind of stability analysis would address a number of fundamental theoretical issues including robustness, accuracy, and reliability of an optimal solution or an equilibrium against errors arising from the problem data or numerical schemes.Let  denote a perturbation of the probability measure .We consider the following perturbed stochastic generalized equations: find a vector  ∈ R  such that 0 ∈ E  [ (, )] + G () . ( Let () and () denote the solution set of (1) and (2), respectively.We investigate the relationship between () and () as  approximates  under some appropriate metric.Shapiro et al. [8] first discussed the sample average approximation (SAA) approach for (1).This method can be seen a special case of the perturbation of  (see Section 4 in this paper).They carried out comprehensive analysis including the existence and convergence of solutions.In our previous work [9], the consistency of Lipschitz-like property of solution map to (1) and its SAA counterpart have been studied.However, the above two studies only focus on the asymptotic analysis and are not related to the quantitative stability analysis.Recently, Liu et al. [7] have studied the qualitative stability of solutions of SGE (1) with  being a set-valued mapping as the underlying probability measure  varies.The results are applied to study the stability of stationary points of several stochastic optimization problems.
In this paper, we follow Liu et al. 's approach to investigate the existence and quantitative stability analysis of solutions of problem (1) when  varies under some appropriate metric.We complement the results of Liu et al. on the issues essentially on twofold: (a) we use Brouwer's fixed point theorem and metric regularity rather than Kummer's results as in [7] to derive conditions for the existence of a solution to perturbation problem (2); (b) by Brouwer's fixed point theorem, we establish the error bounds described by Hausdorff distance between solution sets, instead of the one described by distance from point to set as in [7], to show the quantitative stability of (1) when  varies.We also apply the results to analyze the convergence of SAA method for a class of stochastic conic programming and establish the corresponding error bounds.
This paper is organized as follows: Section 2 gives preliminaries needed throughout the paper.In Section 3, by Brouwer's fixed point theorem, the existence of solution to perturbation problem (2) is investigated.Section 3 provides quantitative stability analysis of problem (1) when  varies under some appropriate metric.In particular, error bounds described by Hausdorff distance between () and () are established.Finally, in Section 4, we apply the results obtained to obtaining quantitative convergence analysis of SAA method for a stochastic conic programming.

Preliminaries
2.1.Notation.Throughout this paper we use the following notations.Let ‖ ⋅ ‖ denote the Euclidean norm of a vector or the Frobenius norm of a matrix and (, ) fl inf   ∈ ‖−  ‖ denote the distance from point  to set .For a multifunction Φ, gphΦ denotes its graph and for a set Ξ, int Ξ denotes its interior.For an extended real-valued function  : R  → R ∪ {±∞}, ∇() denotes the gradient of  at .For a continuously differentiable mapping  : R  → R  , J() denotes the Jacobian matrix of  at .We use B  to denote the closed unite ball in R  , B(, ) the closed ball around  of radius  > 0, and  the identity matrix.For two sets ,  ⊂ R  , we denote by the deviation of set  from the set  and the Hausdorff distance between  and .We use  +  to denote the Minkowski addition of two sets; that is, { +  :  ∈ ,  ∈ }.For the multifunction Φ : R  Î R  , the set lim sup is called the outer limit of Φ as  →  0 .If lim sup → 0 Φ() ⊂ Φ( 0 ), then Φ is outer semicontinuous at  0 (see [10]).It follows from the definition that Φ is outer semicontinuous if and only if gphΦ is closed.

Some Basic Concepts and Results in
see for instance [11].It follows from the definition that the setvalued mapping N  : R  → N  () is outer semicontinuous at  ∈  and If inclusion (8) becomes equality, we say that  is normally regular at  (or Clarke regular by [10]).According to [10, Theorem 6.9], each convex set is normally regular at all its points.
For set-valued maps, the definition of coderivative was introduced by Mordukhovich in [12] based on the limiting normal cone.
Definition 1.Consider a mapping  : R  Î R  and a point  ∈ dom.The coderivative of  at  for any  ∈ () is the mapping  * (, ) : R  Î R  defined by The notation  * (, ) is simplified to  * () when  is single-valued at , () = {}.
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 [11].
(a) (Metric regularity) we say that  −1 is metrically regular at  for  with (, ) ∈ gph −1 if there are some  > 0 and some neighborhoods  of  and  of  such that (b) (Lipschitz-like property) we say that  is Lipschitzlike around (, ) ∈ gph, if there exist some  > 0 and some neighborhoods  of  and  of  such that (c) (Calmness) we say that  is calm at (, ) ∈ gph if there exist some  > 0 and some neighborhoods  of  and  of  such that We know from the definition that the calmness property is weaker than the Lipschitz-like property.As shown in [11], the metrical regularity of  −1 at  for  is equivalent to the Lipschitz-like of  around (, ) ∈ gph, which is equivalent to the coderivative condition This condition is the famous Mordukhovich criterion.

Pseudometric.
Let B denote the sigma algebra of all Borel subsets of Ξ and P be the set of all probability measures of the measurable space (Ξ, B) induced by .To investigate the relationship between the solution sets of SGE ( 2) and (1) when the underlying probability metric  varies, we need to define a metric which is closely related to the involved random functions.Let X be a compact subset of R  .We start by introducing a distance function for the set P, which is appropriate for our problem.Define the set of random functions: The distance function for any probability measures ,  ∈ P is defined by This type of distance was used by Römisch (see [

Main Results
In this section, we discuss existence and quantitative stability analysis of solutions to the perturbed SGE (2).To this end, we assume that X ⊂ R  is a compact and convex set throughout the section and make the following assumptions.

Existence of a Solution to the Perturbed SGE.
We now turn to discuss existence of a solution to the perturbed SGE (2).This issue has been investigated in [7] based on the results of deterministic generalized equations in Kummer [14] under the assumption of convexity of E  [(, )] + G().Without this assumption, we derive the existence results based on the metric regularity.Proof.We know from metric regularity of Ψ  () = E  [(, )]+G() at  for 0 that there exist positive constants , , and  such that for  ∈ B(, ) ∩ X and  ∈ B(0, ).Since Ψ −1  is a convexvalued multifunction, by [ This implies that Υ  () ∈ B(0, ) for  ∈ (,  1 ) and  ∈ .Therefore, by (18), we can define a function for  ∈ (,  1 ).This is a continuous mapping from the compact convex set B(, ) ∩ X to itself.By Brouwer's fixed point theorem, Ĥ has a fixed point in B(, ).We assume that it is   ; then which means that Therefore, and hence () ̸ = 0 for all  close to .Remark 7. We make some comments on the conditions in Corollary 6. (i) The condition that there exists  ∈ () such that  ∈ intX is reasonable in that, by [17, Proposition 2.28] and its proof, () is nonempty if and only if there exists a closed set X with intX ̸ = 0 such that 0 ∈ E  [(, )] + N K∩X () has a solution in intX.In particular, X can be taken to a closed Euclidean ball with the fact that a solution  of () is the center.In this case, we know that  is also a solution of 0 ∈ E  [(, )] + N K ().(ii) In [16, Theorem 3.2], under a calmness condition, condition (25) implies the metrical regularity of Ψ  () at  for 0. In fact, in our case, by Definition 1, such calmness condition holds naturally by the Mordukhovich's criterion.

Stability of SGE.
In this section, we undertake stability analysis of SGE (1), namely, investigating the relationship between the set of solutions to the perturbed SGE (2) and SGE (1) when probability measure  is close to  under the pseudometric defined in the preceding section.
Proof.(i) Notice that () is bounded for  close to .The proof is directly from Theorem 3.1(ii) in [7].Part (ii): since Ψ  is metrically regular at x for 0, there exist positive constants  x,  x, and  x depending on x such that  (,  ()) ≤  x (0, Ψ  ()) (30) for all  ∈ () ∩ B( x,  x).Since  ∈ () means there exists  > 0 such that for  ∈ (), which, by (30), yields for  ∈ () ∩ B( x,  x).Since x is any choice in (), this means that set () may be covered by the union of a collection of -balls; that is, Notice that () is a compact set, by the finite covering theorem.There exist a finite set of points  1 ,  2 , . . .,   ∈ () and a positive number δ > 0 such that In other words, the δ-neighborhood of () may be covered by a finite -net of balls.Let γ = max{  1 , . . .,    }.We have from (33) that  (,  ()) ≤ γD (, ) for all  ∈ () + δB  .Through conclusion (i), we arrive at the fact that (36) holds for all  ∈ () when  is close to , which means the conclusion of (ii).
(iii) We only need to show that there exists  > 0 such that for  close to Step 1.Let x ∈ ().We know from metric regularity of Ψ  at x for 0 that there exist positive constants  x,  x, and  x depending on x such that for all  ∈ B( x,  x) and  ∈ B(0,  x).Since () can be covered by {B( x,  x) : x ∈ ()}, similarly to the proof in (i), there exist  > 0,  > 0 and  > 0 such that for any  ∈ () + B  and  ∈ B(0, ).
Step 2. In particular, for fixed  ∈ (), (39) holds for any  ∈ B(, ) and  ∈ B(0, ).Since Ψ −1  is a convexvalued multifunction, by [ for  close to , which means that Notice that  is an arbitrary choice in () and by (39),  is independent of ; then we know from (43) that for  close to .Combining (28) and (44), we complete the proof of (iii).
Remark 9.In [7], the error bounds described by distance from point to set are obtained to show the qualitative stability analysis of SGE (1).In Theorem 8, we establish the error bounds described by Hausdorff distance between solution sets, which extends the results of [7, Theorem 3.1].
In the case when SGE (1) has a unique solution, we need the concept of strong regularity.Definition 10.Suppose that condition (b) in Assumption 3 holds.We say that a solution  ∈ () ∩ intX is strongly regular if there exist neighborhoods V and W of 0 ∈ R  and , respectively, such that for every  ∈ V, the stochastic generalized equation has a unique solution in W and denoted x = x(), and x(⋅) is Lipschitz continuous on V.
In [18], consistency analysis of strong regular solution of SAA generalized equation is established.For completeness, we present a measure analogue of [ Since as  is close to , then there exists  > 0 such that () has a unique solution in V as  ∈ (, ).This proves (i).The conclusion of (ii) is directly from (48).

Application to Stochastic Conic Programming
Consider the following stochastic optimization problem: where X is a compact and convex subset of R  , K is a closed convex cone in R  ,  : R  × R  → R and  : R  × R  → R  are continuous functions,  : Ω → Ξ is vector of random variables defined on probability space (Ω, F, ) with support set Ξ ⊂ R  , and E  [⋅] denotes mathematical expectation with respect to probability measure.Model (50) can be found in [2] and the linear-quadratic tracking problem is a special case of this model.In this section, we focus on a special case when the probability measure  is approximated by a sequence of empirical measures   defined as where  1 , . . .,   is an independent and identically distributed sampling of  and In this case, where (, ) = ( (,,) (,) ) with (, , ) fl ∇(, ) + J  (, )   and Θ fl X × K − , where K − is the polar cone of K. Similarly, for fixed , if x is the locally optimal solution of problem (54), then under corresponding for any (, ) ∈ , then conclusion (i) in Theorem 8 holds.Notice that by Definition

Conclusion
The existence results and quantitative stability analysis described by Hausdorff distance are established in this paper for SGE (1) when  varies under some appropriate metric, which extends the results in [7].The obtained results are then applied to a stochastic conic programming.In fact, Hausdorff distance type quantitative stability analysis obtained may be applied to more stochastic models such as stochastic mathematical program with equilibrium constraints (SMPEC) and stochastic semi-infinite programming.We let this be our further research topic.