Research Article | Open Access
Stability Analysis of Stochastic Generalized Equation via Brouwer’s Fixed Point Theorem
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 such thatwhere is a continuous function, is a random vector defined on a probability space with support set , denotes the expected value with respect to , and is an outer semicontinuous set-valued mapping. Throughout the paper, we assume that is well defined for any To ease notation, we will use to denote either the random vector or an element of depending on the context.
Model (1) a natural extension of deterministic parametric generalized equation  and the study of stochastic generalized equations can be traced down to King and Rockafellar’s early work . In a particular case when is a normal cone operator in which is a closed convex cone in , (1) reduces to a stochastic variational inequality problem (SVIP) which has been intensively studied over the past few years; see for instance [3–7] 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 such thatLet 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.  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 , 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.  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  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 , 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.
Throughout this paper we use the following notations. Let denote the Euclidean norm of a vector or the Frobenius norm of a matrix and denote the distance from point to set . For a multifunction , denotes its graph and for a set , denotes its interior. For an extended real-valued function , denotes the gradient of at . For a continuously differentiable mapping , denotes the Jacobian matrix of at . We use to denote the closed unite ball in , the closed ball around of radius , and the identity matrix. For two sets , we denote by the deviation of set from the set andthe Hausdorff distance between and . We use to denote the Minkowski addition of two sets; that is, For the multifunction , the set is called the outer limit of as . If , then is outer semicontinuous at (see ). It follows from the definition that is outer semicontinuous if and only if is closed.
2.2. Some Basic Concepts and Results in Set-Valued and Variational Analysis
Given a closed set and a point , the cone is called the Fréchet normal cone to at . Then the limiting normal cone (also known as Mordukhovich normal cone or basic normal cone) to at is defined by see for instance . It follows from the definition that the set-valued mapping is outer semicontinuous at andIf inclusion (8) becomes equality, we say that is normally regular at (or Clarke regular by ). According to [10, Theorem 6.9], each convex set is normally regular at all its points.
If for closed , , then at any with , one has
For set-valued maps, the definition of coderivative was introduced by Mordukhovich in  based on the limiting normal cone.
Definition 1. Consider a mapping and a point . The coderivative of at for any is the mapping 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 .
Definition 2. Consider the multifunction . (a)(Metric regularity) we say that is metrically regular at for with if there are some and some neighborhoods of and of such that (b)(Lipschitz-like property) we say that is Lipschitz-like around , if there exist some and some neighborhoods of and of such that (c)(Calmness) we say that is calm at if there exist some 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 , the metrical regularity of at for is equivalent to the Lipschitz-like of around , which is equivalent to the coderivative conditionThis condition is the famous Mordukhovich criterion.
Let denote the sigma algebra of all Borel subsets of and be the set of all probability measures of the measurable space 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 be a compact subset of . We start by introducing a distance function for the set , which is appropriate for our problem. Define the set of random functions:The distance function for any probability measures is defined by This type of distance was used by Römisch (see [13, Section ]) for the stability analysis of stochastic programming and was called pseudometric in that it satisfies all properties of a metric except that does not necessarily imply unless the set of functions is sufficiently large. It is well known that is nonnegative and symmetric and satisfies the triangle inequality; see [13, Section ].
3. 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 is a compact and convex set throughout the section and make the following assumptions.
Assumption 3. Let be a set of probability measures such that . The following hold: (a) is continuous at for each and there exists satisfying such that for each and .(b) is continuously differentiable at for each and there exists satisfying such that for each and .
We know from [8, Theorem 7.43, ] that condition (a) in Assumption 3 means that for , is well defined and continuous at each and under condition (b) in Assumption 3, is continuously differentiable and .
3.1. 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  based on the results of deterministic generalized equations in Kummer  under the assumption of convexity of . Without this assumption, we derive the existence results based on the metric regularity.
Theorem 4. Suppose that and there exist such that . If condition (a) in Assumption 3 holds and (a) is metrically regular at for and (b) is a convex-valued multifunction, then for all close to .
Proof. We know from metric regularity of at for that there exist positive constants , , and such that for and . Since is a convex-valued multifunction, by [15, Theorem 2.2], we can define a continuous function such that for andFor each , let and . Since under condition (a) in Assumption 3,as is close to , there exists such that This implies that for and . Therefore, by (18), we can define a functionfor . This is a continuous mapping from the compact convex set to itself. By Brouwer’s fixed point theorem, has a fixed point in . We assume that it is ; then which means that Therefore, and hence for all close to .
Remark 5. In the case when in (1) with being a closed convex set, we may have more simple results; that is, if , then and for close to .
Corollary 6. Suppose that in (1) with being a closed convex set and being a normal cone operator and . Suppose that there exist such that and Assumption 3 holds. If is monotone on and conditionholds, then for close to .
Proof. We only need to verify the conditions in Theorem 4. We know from [16, Theorem 3.2] that if condition (b) in Assumption 3 holds, then under condition (25), is metrically regular at for . This verifies condition (a) in Theorem 4. Since is monotone on , we have for , which means that is monotone on for any . Then is monotone on , which, by [17, Theorem 2.3.5] and condition (a) in Assumption 3, means that is a convex-valued mapping; this verifies condition (b) in Theorem 4.
Remark 7. We make some comments on the conditions in Corollary 6. (i) The condition that there exists such that is reasonable in that, by [17, Proposition 2.28] and its proof, is nonempty if and only if there exists a closed set with such that has a solution in . In particular, 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 . (ii) In [16, Theorem 3.2], under a calmness condition, condition (25) implies the metrical regularity of at for . In fact, in our case, by Definition 1, such calmness condition holds naturally by the Mordukhovich’s criterion.
3.2. 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.
Theorem 8. Suppose that , for close to , and condition (a) in Assumption 3 holds. If is a convex compact set such that and for close to , then the following assertions hold:(i)For any small positive number , there exists such that for any , where .(ii)If is metrically regular at any for 0, then there exists such that for any (iii)If is metrically regular at any for and is a convex-valued multifunction, then there exists such that for any
Proof. (i) Notice that is bounded for close to . The proof is directly from Theorem 3.1(ii) in .
Part (ii): since is metrically regular at for , there exist positive constants , , and depending on such thatfor all . Since means there exists such that for , which, by (30), yieldsfor . Since 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 and a positive number such that In other words, the -neighborhood of may be covered by a finite -net of balls. Let . We have from (33) thatfor all . 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 such that for close to .
Step 1. Let . We know from metric regularity of at for that there exist positive constants , , and depending on such thatfor all and . Since can be covered by , similarly to the proof in (i), there exist , and such thatfor any and .
Step 2. In particular, for fixed , (39) holds for any and . Since is a convex-valued multifunction, by [15, Theorem 2.2], we can define a continuous function such that for andFor each , let and . Under condition (a) in Assumption 3, there exists such that which means that for close to . Let . We know from (40) that is a continuous function which maps to itself uniformly for close to . By Brouwer’s fixed point theorem, for each , there exist such that ; that is, and by (40), for close to , which means thatNotice that is an arbitrary choice in and by (39), is independent of ; then we know from (43) thatfor close to . Combining (28) and (44), we complete the proof of (iii).
Remark 9. In , 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 is strongly regular if there exist neighborhoods and of and , respectively, such that for every , the stochastic generalized equationhas a unique solution in and denoted , and is Lipschitz continuous on .
Theorem 11. Suppose that condition (b) in Assumption 3 holds. If there exists such that and is strongly regular, then(i)there exists such that for every , contains a unique solution, denoted by ;(ii)there exist constants and such that for all .
Proof. Let be a convex compact neighborhood of and be a space of continuously differentiable mappings equipped with the norm Since is a strongly regular solution of the stochastic generalized equation, there exists such that for any satisfying , has a unique solution such that is Lipschitz continuous with a Lipschitz modular, denoted by with respect to the norms andSinceas is close to , then there exists such that has a unique solution in as . This proves (i). The conclusion of (ii) is directly from (48).
4. Application to Stochastic Conic Programming
Consider the following stochastic optimization problem:where is a compact and convex subset of , is a closed convex cone in , and are continuous functions, is vector of random variables defined on probability space with support set , and denotes mathematical expectation with respect to probability measure. Model (50) can be found in  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 is an independent and identically distributed sampling of and In this case,By classical law of large numbers in statistics, and converge to and , respectively, as tends to infinity. This kind of approximation is known in stochastic programming under various names such as sample average approximation (SAA), Monte Carlo method, and sample path optimization; see [4, 8] and the references therein. Consequently, by the SAA method, problem (50) can be approximated by the following problem:Problem (54) is called the SAA problem and (50) the true problem.
If condition (b) in Assumption 3 holds for mappings and , respectively, is convex with respect to for each , that is, the multifunction is convex for each , and the constraint qualificationholds, then by [19, Proposition 2.104, Theorem 3.9], for any locally optimal solution of , there exists such that satisfies the following stationary condition:which can be rewritten as a stochastic generalized equation where with and , where is the polar cone of . Similarly, for fixed , if is the locally optimal solution of problem (54), then under corresponding constraint qualification, there exists multiplier such that satisfies the following stationary condition: where with and .
Let and Next we show that all the two sets are bounded.
Lemma 12. If condition (b) in Assumption 3 holds for mappings and , respectively, is convex with respect to for each , and condition (55) holds, then there exist a compact set and a number such that almost surely.
Proof. We at first show that there exists a number such that is bounded almost surely. Assume by contradiction that there exists a sequence satisfying and with probability one (w.p.1) as . Then there exist a sequence and a number sequence such thatwith and w.p.1 as . Under condition (b) in Assumption 3, we have by the law of large numbers that , , and converge to , , and w.p.1, respectively. Since is a compact set, without loss of generality, we may assume that w.p.1 as tends to infinity. Then by the outer semicontinuity of normal cone, it holds that Multiplying by the two sides of (59) and letting tend to infinity, we obtainwith . Since condition (55) holds, we have from [19, Proposition 2.104] that holds, which, by [20, Proposition 2.2], is equivalent to This, by (61), means that , which is a contradiction. Therefore there exists a number such that is bounded almost surely. In the similar way, we can demonstrate that is bounded.
Let and We want to demonstrate the quantitative stability of when it is approximated by .
Theorem 13. If condition (b) in Assumption 3 holds for mappings , respectively, is convex with respect to for each and condition (55) holds. If conditionholds for any , then the following hold:(i)There exists such that for is large enough, where .(ii)If, in addition, the matrix is semidefinite for any , then there exists such that
for large enough.
Proof. It suffices to verify the conditions in Theorem 8. By Lemma 12, there exist a compact set and a number such that almost surely, which means that contains in a compact set almost surely. We know from [16, Theorem 3.2] that iffor any , then conclusion (i) in Theorem 8 holds. Notice that by Definition 1, and is matrix (66), which implies that (67) is equivalent to condition (64). Therefore by Theorem 8, conclusion (i) holds.
We know from [17, Proposition 2.3.2] that if matrix (66) is semidefinite, then is monotone, which, by the proof of Corollary 6, means that is a convex-valued mapping; this verifies condition in (ii) of Theorem 8. Conclusion (ii) follows from Theorem 8.
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 . 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.
Conflicts of Interest
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.
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