eu CONSISTENCY ANALYSIS OF AUBIN PROPERTY OF SAA SOLUTION MAPPING FOR A STOCHASTIC COMPLEMENTARITY PROBLEM

In this paper, we investigate properties of sample average approximation (SAA) solution mapping for a parametric stochastic complementarity problem, where the underlying function is the expected value of stochastic function. In particular, using the notion of cosmic deviation, which is originated from the concept of cosmic distance in variational analysis, we develop sufficient conditions for the consistency of Aubin property of the solution mapping of the SAA parametric stochastic complementarity problems, namely if the solution map of the true problem has the Aubin property around some point, then so does the SAA problem around reference point with probability one when the sample size is large enough. At last, an example is illustrated to show the application of the analysis. AMS Subject Classification: 90C30


Introduction
Our concern in this paper is the following parametric stochastic complementarity problem (PSCP) Ψ(x, y) ≥ 0, y ≥ 0, Ψ(x, y) T y = 0, Ψ(x, y) = E[F (x, y, ξ(ω))] (1) where F : ℜ n × ℜ m × ℜ k → ℜ m is a random mapping, ξ : Ω → Ξ ⊂ ℜ k is a random vector defined on probability space (Ω, F, P), E denotes the mathematical expectation.Throughout the paper, we assume that E[F (x, y, ξ(ω))] is well defined and finite for any x ∈ ℜ n and y ∈ ℜ m .To ease the notation, we write ξ(ω) as ξ and this should be distinguished from ξ being a deterministic vector of Ξ in a context.PSCP is a natural extension of deterministic complementarity problem, which contain some expectation of uncertain factors.When the parameter is fixed, the PSCP is the stochastic complementarity problem (SCP), some examples of SCP arising from the areas of economics engineering and operations management, can be found in [3] and [2].Exact evaluation of the expected value is impossible or prohibitively expensive, see [3].Many authors have suggested the sample average approximation (SAA) method to solve SCP, see for example [7], [8], [9], [10], [11].The basic idea of SAA is to generate an independent identically distributed (iid) sample ξ 1 , • • • , ξ N of ξ and then approximate the expected value with sample average.In this context, PSCP (1) is approximated by where is the sample-average mapping of F (x, y, ξ).We refer to (1) as the true problem and (2) as the SAA problem to (1).Let be the solution set of SCP for a given x ∈ ℜ n and S N (x) the solution set of SAA problem.When using SAA method, an important issue is that if the true problem has nice property such as Aubin property, whether or not the SAA problem do as the sample size large enough?To answer this question, we need study the consistency of Aubin property of SAA solution mapping for SCP (1).
The purpose of this paper is to obtain sufficient conditions for the consistency of Aubin property of solution map of SAA estimator (2).The main tool, used for the analysis, is the cosmic deviation, which is originated from cosmic distance in [6,Section 4] and used to measure the convergence of sequence of unbounded sets.The reason is that the coderivatives, which is closely related to the Aubin property, are unbounded and the usual deviation defined by Hausdorff distance is not suitable.
This paper is organized as follows: Section 2 gives preliminaries needed throughout the whole paper.In section 3, we analyze the consistency of the Aubin property of solution map of SAA problem.

Preliminaries
Throughout this paper we use the following notations.Let • denote the Euclidean norm of a vector or the Frobenius norm of a matrix and d(x, D) := inf x ′ ∈D x − x ′ denote the distance from point x to set D. For a m × n matrix A, A ij denotes the the element of the i-th row and j-th column of A. For a multifunction Φ, gph Φ denotes its graph and for a set Ω, int Ω, cl Ω denotes its interior, closure hull, respectively.When Ω = ∅, Ω ∞ denotes the horizon cone of Ω defined by ( [6]) Ω ∞ = {x : ∃x ν ∈ Ω, λ ν ց 0, with λ ν x ν → x}.We use B to denote the closed unite ball, B δ (x) the closed ball around x of radius δ > 0 and I the identity matrix.For a continuously differentiable map F : ℜ n → ℜ m , J F (z) denotes the Jacobian of F at z ∈ ℜ n .
For two sets A, C ⊂ ℜ n , we denote by D(A, C) := inf{t > 0 : A ⊂ C + t B} the deviation of set A from the set C. The limiting (Mordukhovich) normal cone to set Ξ at x ∈ Ξ is defined by where and "lim sup" denotes outer limit of set-valued mapping or upper limit of realvalued mapping, see [6].
For set-valued maps, the definition of coderivative [6, definition 8.33] is established with the help of the definition for limiting normal cone.Definition 1.Consider a mapping S : ℜ n ⇒ ℜ m and a point x ∈ domS.The coderivative of S at x for any ū ∈ S(x) is the mapping D * S(x, ū) : ℜ m ⇒ ℜ n defined by The Aubin property is as follows, see [1].
Definition 2. Consider the multifunction F : ℜ m ⇒ ℜ n .We say that F has Aubin property around (ȳ, x) ∈ gphF , if there exist some κ > 0 along with some neighborhoods U of x and V of ȳ with This condition is the famous Mordukhovich criterion ([6, Theorem 9.40]).
To measure the convergence of a sequence of unbounded sets, we need the following concept, which can be found in [10].The cosmic deviation has the following properties.
Proposition 4. [12, Proposition 2.3] For C ν and C in ℜ n ,the following statements are equivalent:

Main Results
In this section, we derive conditions ensuring the consistency of Aubin property of solution mapping of SAA estimator (2).
In what follows, we make the following assumptions to make (1) more clearly defined and to facilitate the analysis.
Assumption 1 There exists a measurable function κ(ξ) such that Assumption 3 There exists an nonnegative measurable function g(ξ) such that E[g(ξ)] < +∞ and We at first provide some lemmas. where where the index sets with the pair (a, b) ∈ gphN ℜ m + .The following lemma estimates the coderivatives of the solution maps to the SAA parametric complementarity problem (2).Lemma 6.Let x ∈ ℜ n be fixed, ȳ ∈ S(x) and ȳN ∈ S N (x) w.p.1.Suppose Assumptions 1-3 are satisfied.(a) If the following qualification condition holds:

Then we have for all y
where (b) If the following qualification condition holds: where L := L(ȳ, −Ψ(x, ȳ)), where Proof.For (a), the solution set to (2) can be rewritten as where which means gph S N can be rewritten as Then we know from [6, Theorem 6.14] that if the following qualification condition holds: Then we obtain where u L = 0 and for i ∈ Î0 either u i v i = 0 or u i < 0 and v i > 0 w.p.1.Noticing that by Lemma 5, ( 8) is just ( 4) and (10) implies that x * N ∈ A N (x, ȳN , y * ) w.p.1.. Therefore the inclusion (5) holds.We complete the proof of (a).The assertion (b) can be demonstrated similarly.
Next we show the consistency of Aubin property of SAA problem to that of corresponding true problem.4) and ( 6) in Lemma 6 hold .
Proof.We proceed the proof in two steps.For z * N , there exists λ N > 0 and We know from the proof of Lemma 6 that there exists Next we demonstrate that if z * N → z * almost surely as N → ∞, then the sequence {η N } is bounded almost surely.Assume by contradiction that η N → ∞ w.p.1 as N → ∞, then there exists α N ց 0 such that almost surely.By multiplying α N to both sides of ( 12), we obtain By Lemma 7, we have Letting N → ∞ in (14), we have from ( 13) and ( 15) We know from Lemma 5 that condition (11) is equivalent to which, together with ( 16) and (17), implies that x * = 0 w.p.1.Then, by Lemma 5, let η = ( u, v), we know from (16) that with ũL = 0, ṽI + = 0 and for i ∈ I 0 either ũi ṽi = 0 or ũi < 0 and v i > 0, which, by condition (6), means η = 0 w.p.1.This leads to a contradiction.Hence {η N } is almost surely bounded.