JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 321010 10.1155/2014/321010 321010 Research Article A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints Li Yu-xin 1 Zhang Jie 2 Xia Zun-quan 1 Cen Song 1 Institute of ORCT School of Mathematical Sciences Dalian University of Technology Dalian 116024 China dlut.edu.cn 2 School of Mathematics Liaoning Normal University Dalian 116029 China lnnu.edu.cn 2014 1022014 2014 29 05 2013 03 11 2013 10 2 2014 2014 Copyright © 2014 Yu-xin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

1. Introduction

Our concern in this paper is the following stochastic mathematical program with complementarity constraints (SMPCC): (1)min𝔼[f(z,ξ(ω))]s.t.𝔼[g(z,ξ(ω))]0,s.t.𝔼[h(z,ξ(ω))]=0,s.t.0𝔼[G(z,ξ(ω))]s.t.𝔼[H(z,ξ(ω))]0, where f:n×k, g:n×kp, h:n×kq, H:n×km, and G:n×km are random mappings; ξ:ΩΞk is a random vector defined on a probability space (Ω,,P); 𝔼 denotes the mathematical expectation; the notation means “perpendicular.” Throughout the paper, we assume that 𝔼[f(z,ξ(ω))], 𝔼[g(z,ξ(ω))], 𝔼[h(z,ξ(ω))], 𝔼[G(z,ξ(ω))], and 𝔼[H(z,ξ(ω))] are all well defined and finite for any zn. 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.

The SMPCC (1) is a natural extension of deterministic mathematical program with complementarity constraints (MPCC) [1, 2], which have many applications in transportation  and communication networks , and so forth. There are many stochastic formulations of MPCC proposed in the recent years [3, 57]. Among these formulations, Birbil et al.  applied sample path method  to SMPCC (1).

In this paper, we are concerned with a numerical method for solving (1). Evidently, if the integral involved in the mathematical expectation of problem (1) can be evaluated either analytically or numerically, then problem (1) can be regarded as the usual MPCC problem and consequently it can be solved by existing numerical methods that are related. However, as shown in , in many situations, exact evaluation of the expected value in (1) for x is either impossible or prohibitively expensive. Sample average approximation (SAA) method [8, 10] is suggested by many authors to handle such difficulty; see the recent works . The basic idea of SAA is to generate an independent identically distributed (iid) sample ξ1,,ξN of ξ and then approximate the expected value with sample average. In this context, let ξ1,,ξN be iid sample; then the SMPCC (1) is approximated by the following SAA problem: (2)minf^N(z)s.t.g^N(z)0,h^N(z)=0,dddd.0G^N(z)H^N(z)0, where f^N(z):=(1/N)i=1Nf(z,ξi), g^N(z):=(1/N)i=1Ng(z,ξi), h^N(z):=(1/N)i=1Nh(z,ξi), G^N(z):=(1/N)i=1NG(z,ξi), H^N(z):=(1/N)i=1NH(z,ξi) is the sample-average function of f(z,ξi), g(z,ξi), h(z,ξi), G(z,ξi) and H(z,ξi) respectively. We refer to (1) as the true problem and (2) as the SAA problem to (1). Another critical problem for solving (1) is how to solve SAA problem (2) effectively. Since the Mangasarian-Fromovitz constraint qualification is violated at every feasible point of SAA problem (2) (see ), it is not appropriate to use standard nonlinear programming software to solve the SAA problem directly. The well-known regularization scheme , is a effective way to deal with this issue. That is, by replacing the complementarity constraint with a parameterized system of inequalities, the SAA problem is reformulated as follows: (3)minf^N(z)s.t.g^N(z)0,h^N(z)=0,s.t.G^N(z)0,H^N(z)0,s.t.G^N(z)H^N(z)tNe, where tN>0 is a parameter, “” denotes the Hadamard product and em is a vector with components 1. Then the SAA problem can be approximated by a smooth nonlinear programming (NLP) problem (3) when the parameter is sufficiently small. Consequently, a solution to true problem (1) can be obtained by solving a sequence of such regularized SAA problems.

In this paper, we focus on the detailed analysis of convergence properties of the regularized SAA problem (3) to the true problem (1) as the sample size tends to infinity. The main contributions of this paper can be summarized as follows: by the notion of epiconvergence in , we establish the almost sure convergence of optimal solutions of smoothed SAA problem as the sample size tends to infinity. Under MPCC-MFCQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is a kind of stationary point almost surely. The obtained results can be seen an improvement of [17, Theorem 3.1] for solving SMPCC under weaker constraint qualification conditions. Moreover, under the MPCC strong second-order sufficient condition (MPCC-SSOSC) in , we investigate sufficient conditions under which the smoothed SAA problem possesses a Karash-Kuhn-Tucker point when the sample size is large enough, and the sequence of those points converges exponentially to a kind of stationary point of SMPCC almost surely as the sample size tends to infinity.

This paper is organized as follows: Section 2 gives preliminaries needed throughout the whole paper. In Sections 3 and 4, we establish the almost sure convergence of optimal solutions and stationary points of the regularized SAA problem as the sample size tends to infinity respectively. In Section 5, existence and exponential convergence rate of stationary points of the regularized SAA problem are investigated. We also report some preliminary numerical results in Section 6.

2. Preliminaries

Throughout this paper we use the following notations. Let · denote the Euclidean norm of a vector or the Frobenius norm of a matrix. For a m×n matrix A, Aij denotes the element of the ith row and jth column of A. We use In to denote the n×n identity matrix, 𝔹 denotes the closed unite ball, and 𝔹(x,δ) denotes the closed ball around x of radius δ>0. For a extended real-valued function φ:n{±}, epiφ, φ(x), and 2φ(x) denote their epigraph that is, the set {(x,α)φ(x)α}, the gradient of φ at x, and the Hessian matrix of φ at x, respectively. For a mapping ϕ:nm, 𝒥ϕ(x) denotes the Jacobian of ϕ at x. ++ stands for the positive real numbers.

In the following, we introduce some concepts of the convergence of set sequences and mapping sequences in  which will be used in the next section. Define (4)𝒩{NNNNfinite},𝒩#{NNNinfinite}, where N denotes the set of all positive integer numbers.

Definition 1.

For sets Cν and C in n with C closed, the sequence {Cν}νN is said to converge to C (written CνC) if (5)limsupνCνCliminfνCν with (6)limsupνCν:={xνNxx  N𝒩#,limsupνCν:{=  xνCν(νN)  such  that  xνNx},liminfνCν:={xνNxx  N𝒩,limsupνCν:{=  xνCν(νN)  such  that  xνNx}.

The continuous properties of a set-valued mapping S can be developed by the convergence of sets.

Definition 2.

A set-valued mapping S:nm is continuous at x-, symbolized by limxx-S(x)=S(x-), if (7)limsupxx-S(x)S(x-)liminfxx-S(x).

Definition 3.

Consider now a family of functions fν:n-, where ¯={±}. One says that fν epiconverges to a function f:n¯ as ν and is written as (8)f=e-limνfν, if the sequence of sets epi fν converges to epi f in n× as ν.

Definition 4.

Given a clos set Ξn and a point x¯Ξ. The cone (9)N^Ξ(x¯):={x*nlimsupxΞx¯x*,x-x¯x-x¯0} is called the Fréchet normal cone to Ξ at x¯. Then the limiting normal cone (also known as Mordukhovich normal cone or basic normal cone) to Ξ at x¯ is defined by (10)NΞ(x¯):=limsupxΞx¯N^Ξ(x). If Ξn is a closed convex set, the limiting normal cone NΞ(x¯) is the normal cone in the sense of convex analysis.

Next, we recall some basic concepts that are often employed in the literature on optimization problems with complementarity constraints.

Let z¯ be a feasible point of problem (1) and for convenience we define the index sets (11)Ig={i{1,2,,p}:𝔼[gi(z¯,ξ)]=0},α¯={i{1,2,,m}:𝔼[Gi(z¯,ξ)]=0<𝔼[Hi(z¯,ξ)]},β¯={i{1,2,,m}:𝔼[Gi(z¯,ξ)]=0=𝔼[Hi(z¯,ξ)]},γ¯={i{1,2,,m}:𝔼[Gi(z¯,ξ)]>0=𝔼[Hi(z¯,ξ)]}.

The constraint qualifications for SMPCC is as follows.

Definition 5.

Assume 𝔼[g(·,ξ)], 𝔼[h(·,ξ)], 𝔼[G(·,ξ)], and 𝔼[H(·,ξ)] are continuously differentiable at z¯. We say the MPCC Mangasarian-Fromovitz constraint qualification (MPCC-MFCQ) holds at z¯ if the set of vectors (12){α¯β¯;𝔼[Hi(z¯,ξ)]𝔼[hi(z¯,ξ)],i=1,,q;𝔼[Gi(z¯,ξ)],iα¯β¯;𝔼[Hi(z¯,ξ)],iβ¯γ¯} are linearly independent and there exists a nonzero vector dn such that (13)𝔼[hi(z¯,ξ)]Td=0i=1,,q,𝔼[Gi(z¯,ξ)]Td=0iα¯β¯,𝔼[Hi(z¯,ξ)]Td=0iγ¯β¯,𝔼[gi(z¯,ξ)]Td<0iIg.

Definition 6.

Assume 𝔼[g(·,ξ)], 𝔼[h(·,ξ)], 𝔼[G(·,ξ)], and 𝔼[H(·,ξ)] are continuously differentiable at z¯. We say the MPCC linear independence constraint qualification (MPCC-LICQ) holds at z¯ if the set of vectors (14){𝔼[gi(z¯,ξ)],iIg;𝔼[hi(z¯,ξ)],i=1,,q,𝔼[Gi(z¯,ξ)],iα¯β¯;𝔼[Hi(z¯,ξ)],iβ¯γ¯} are linearly independent.

As in , we use the following two stationarity concepts for SMPCC.

Definition 7.

Assume z¯ is a feasible point of SMPCC (1), 𝔼[g(·,ξ)], 𝔼[h(·,ξ)], 𝔼[G(·,ξ)], and 𝔼[H(·,ξ)] are continuously differentiable at z¯. Suppose there exist vectors λ¯|Ig|, μ¯q, u¯|α|+|β|, and v¯|β|+|γ| such that z¯ satisfies the following conditions: (15)0=𝔼[f(z¯,ξ(ω))]+iIgλ¯i𝔼[gi(z¯,ξ)]+i=1qμ¯i𝔼[hi(z¯,ξ)]-iα¯β¯u¯i𝔼[Gi(z¯,ξ)]-iβ¯γ¯v¯i𝔼[Hi(z¯,ξ)].

(C-stationary point) We call z¯ a Clarke stationary point of (1) if u¯iv¯i0, iβ¯.

(S-stationary point) We call z¯ a strongly stationary point of (1) if u¯i0, v¯i0, iβ¯.

The following upper level strict complementarity condition was used in  in the context of sensitivity analysis for MPCC.

Definition 8.

We say that the upper level strict complementarity condition (ULSC) holds at z¯ if u¯i and v¯i, the multipliers correspondence to 𝔼[Gi(z¯,ξ)], and 𝔼[Hi(z¯,ξ)], respectively, satisfy u¯iv¯i0 for all iβ¯.

It is well known that a point (x¯,y¯) satisfies the lower level strict complementarity condition (LLSC) if 𝔼[Gi(z¯,ξ)]+𝔼[Hi(z¯,ξ)]>0 hold for all i{1,,m}, we can see from an example from  that ULSC condition is considerably weaker than the LLSC condition, and in practice, it may make more sense than the latter one.

We use the following second-order condition based on the MPCC-Lagrangian: (16)L(z,λ,μ,u,v)=𝔼[f(z,ξ)]+iIgλi𝔼[gi(z,ξ)]+i=1qμi𝔼[hi(z,ξ)]-iα¯β¯ui𝔼[Gi(z,ξ)]-iβ¯γ¯vi𝔼[Hi(z,ξ)] of (P).

Definition 9 (see [<xref ref-type="bibr" rid="B16">16</xref>]).

Let z¯ be a S-stationary point of (1) and (λ¯,μ¯,u¯,v¯) is the corresponding multiplier at z¯. Suppose 𝔼[g(·,ξ)], 𝔼[h(·,ξ)], 𝔼[G(·,ξ)], 𝔼[H(·,ξ)], and 𝔼[f(·,ξ)] are twice continuously differentiable at z¯. We say that the MPCC strong second-order sufficient condition (MPCC-SSOSC) holds at z¯ if (17)dTz2L(z¯,λ¯,μ¯,u¯,v¯)d>0 for every nonvanishing d with (18)𝔼[gi(z¯,ξ)]Td=0,iIg,𝔼[hi(z¯,ξ)]Td=0,i=1,2,,q,𝔼[Gi(z¯,ξ)]Td=0,iα¯,𝔼[Hi(z¯,ξ)]Td=0,iγ¯,min{𝔼[Gi(z¯,ξ)]Td,𝔼[Hi(z¯,ξ)]Td}=0,min{𝔼[Gi(z¯,ξ)]Td,𝔼[Hi(z¯,ξ)]T}iβ¯. Assume z- is a S-stationary point of (1) and (λ-,μ-,u-,v-) is the corresponding multiplier. Then we know from [16, Theorem 7] that if MPCC-SSOSC holds at z-, it is a strict local minimizer of the SMPCC (1).

Throughout the paper, we assume the sample ξ1,,ξN of the random vector ξ is iid and give the following assumptions to make (1) more clearly defined and to facilitate the analysis.

Assumption 10.

The mapping f(·,ξ), G(·,ξ), H(·,ξ), g(·,ξ), and h(·,ξ) are twice continuously differentiable on n a.e. ξΞ.

Assumption 11.

For any z-n, there exists a closed bounded neighborhood D of z- and a nonnegative measurable function κ(ξ) such that 𝔼[κ(ξ)]<+ and (19)supzDmax{ϱ(z,ξ)}κ(ξ) for all ξΞ, where ϱ(z,ξ) is any element in the collection of functions {f(z,ξ), g(z,ξ), h(z,ξ), G(z,ξ), H(z,ξ), 𝒥z  g(z,ξ), z  f(z,ξ), 𝒥z  h(z,ξ), 𝒥z  G(z,ξ), and 𝒥z  H(z,ξ)}.

Assumption 12.

For every i{1,,n}, the following properties hold ture.

For every zn, the moment generating function (20)(t)i:=𝔼[e([zf(z,ξ)]i-[𝔼(zf(z,ξ))]i)]

of random variable [zf(z,ξ)]i-[𝔼(zf(z,ξ))]i is finite valued for all t in a neighborhood of zero.

There exists a measurable function κ:Ξ+ such that (21)[z  f(z,ξ)]i-[zf(z,ξ)]iκ(ξ)z-z

for all ξΞ and z,zn.

The moment generating κ(t)=𝔼[etκ(ξ)] of κ(ξ) is finite valued for all t in a neighborhood of zero.

Assumptions 1012 are popularly used conditions for the analysis of SAA method for stochastic programming. Under Assumptions 1011, we know from [10, Chapter 7], that 𝔼[f(z,ξ(ω))] and 𝔼[G(z,ξ(ω))] are twice continuously differentiable on n. In particular, (22)𝔼[f(z,ξ(ω))]=𝔼[f(z,ξ(ω))],𝒥𝔼[G(z,ξ(ω))]=𝔼[𝒥G(z,ξ(ω))]. Assumption 12 is used to ensure exponential convergence rate of proposed regularization SAA method in Section 5.

The following results are directly from the Uniform Laws of Large Numbers in [10, Theorem 7.48].

Lemma 13.

Let z¯ be a feasible point of (1). Suppose that Assumptions 1011 are satisfied; then we obtain (23)supzD1Ni=1Nϱ(z,ξ)-𝔼[ϱ(z,ξ)]0w.p.1, where the set D is a closed bounded neighborhood of z¯ and ϱ(z,ξ) is any element in the collection of functions {f(z,ξ), g(z,ξ), h(z,ξ), G(z,ξ), H(z,ξ), 𝒥zg(z,ξ), zf(z,ξ), 𝒥zh(z,ξ), 𝒥zG(z,ξ), and 𝒥zH(z,ξ)}.

3. Almost Sure Convergence of Optimal Solutions

In this section, by the notion of epiconvergence in , we establish the almost convergence of optimal solutions of regularized SAA problem (3) to those of SMPCC (1) as the sample size tends to infinity.

Let us introduce some notions: (24)0:={zn:𝔼[g(z,ξ)]0,𝔼[h(z,ξ)]=0,1111110𝔼[G(z,ξ)]𝔼[H(z,ξ)]0},N:={zn:g^N(z,ξ)0,h^N(z,ξ)=0,111111G^N(z,ξ)H^N(z,ξ)tNe},f¯N(z):=f^N(z)+δN(z),f¯(z):=𝔼[f(z,ξ)]+δ0(z),κ0:=inf{𝔼[f(z,ξ)]:z0},S0=argmin{𝔼[f(z,ξ)]:z0},SN:=argmin{f^N(z):zN}.

Now we give a conclusion about the almost sure convergence of the set N as N tends to infinity in the following proposition.

Proposition 14.

Let tN0 as N. Suppose Assumptions 1011 hold. If MPCC-LICQ (Definition 6) holds for any z0, then (25)limNN=0w.p.1.

Proof.

We at first show that limsupNN0 w.p.1. It suffices to prove that for a sequence {zN} satisfying zNN for each N, if zN converges to z¯ w.p.1 as N, then z¯0 w.p.1. Indeed, we know from the definition of N that zN satisfies (26)g^iN(zN)0,h^jN(zN)=0,H^kN(zN)0,G^kN(zN)0,H^kN(zN)G^kN(zN)tN for i=1,2,,p;j=1,2,,q; and k=1,2,,m, which, by Lemma 13, means that z¯0 w.p.1.

Let z¯0. Next we show that z¯liminfNN w.p.1. Let (27)Σ(z)={zn:Ψ(z)-p=0}, where the mapping Ψ:n2m+p+|Ig| is defined by (28)Ψ(p)=(𝔼[gi(z,ξ)],iIg𝔼[hi(z,ξ)],i=1,2,,p𝔼[Gi(z,ξ)],iα¯β¯𝔼[Hi(z,ξ)],iγ¯β¯). Then z¯Σ(0)Q00, where (29)Q0={zn:𝔼[Hi(z,ξ)]>0,iα¯𝔼[Gi(z,ξ)]>0,iγ¯𝔼[gi(z,ξ)]>0,iIg}. Under MPCC-LICQ, Σ(·) has Aubin property  around (0,z¯), which means that there exist constants c>0,ɛ>0, and δ>0 such that (30)dist(z,Σ(p))cdist(p,Σ-1(z)) holds for z𝔹(z¯,ɛ) and p𝔹(0,δ). Therefore, for sufficiently small positive numbers ɛ,δ, there exists a continuous function z(·):𝔹(0,δ)𝔹(z¯,ɛ) such that z(0)=z¯ and for any p𝔹(0,δ), (31)Ψ(z(p))-p=0. Let (32)pN(z)=(𝔼[gi(z,ξ)]-g^iN(z),iIg𝔼[h(z,ξ)]-h^N(z)𝔼[Gi(z,ξ)]-G^iN(z),iα¯𝔼[Gi(z,ξ)]-G^iN(z)+tN,iβ¯𝔼[Hi(z,ξ)]-H^iN(z),iγ¯𝔼[Hi(z,ξ)]-H^iN(z)+tN,iβ¯). Then, by Lemma 13, we have for N large enough (33)maxz𝔹(z¯,ɛ)pN(z)<δw.p.1 and for any z𝔹(z¯,ɛ), (34)z(pN(z))-z¯ɛw.p.1. Define a function (35)φN:𝔹(z¯,ɛ)𝔹(z¯,ɛ)zz(pN(z)). This is a continuous mapping from the compact convex set 𝔹(z¯,ɛ) to itself. By Brouwer's fixed theorem, φN has a fixed point. Hence, there exists a vector zN𝔹(z¯,ɛ) w.p.1 such that zN=φN(zN)=z(pN(zN)). Therefore, we have from (31) that (36)0=Ψ(z(pN(zN)))-pN(zN). That is, zN0N,w.p.1, where (37)0N={zn:g^iN(z)=0,iIgh^N(z)=0G^iN(z)=0,iα¯G^iN(z)=tN,iβ¯H^iN(z)=0,iγ¯H^iN(z)=tN,iβ¯}. By Lemma 13, we obtain for sufficiently large N, zNQN due to z¯Q0, where (38)QN={zn:H^iN(z)>0,iα¯G^iN(z)>0,iγ¯g^iN(z)>0,iIg}, which means that zNQN0NN. As a result, z¯ belongs to liminfNN w.p.1 because of the almost sure convergence of zN to z¯ as N. We complete the proof.

By Definition 3, similarly to the proof of [15, Lemma 4.3], we obtain the following lemma.

Lemma 15.

Under the conditions of Proposition 14, we have (39)e-limNf¯N=f¯w.p.1.

The following result is directly from [18, Theorem 7.31].

Theorem 16.

Suppose zN solves (3) for each N and z¯ is almost surely an accumulate point of the sequence {zN}. If the conditions in Proposition 14 hold and κ0 is finite, then z¯ is almost surely an optimal solution of the true problem (1).

4. Almost Sure Convergence of Stationary Points

In practice, finding a global minimizer might be difficult and in some cases we might just find a stationary point. As a result, we want to know whether or not an accumulation point of the sequence of stationary points of regularized SAA problem (3) is almost surely a kind of stationary point of SMPCC (1).

Notice that (3) is a standard nonlinear programming with smooth constraints. If zN is a local optimal solution of the regularized SAA problem (3), then under some constraint qualifications, zN is a stationary point of (3); namely, there exists Lagrange multipliers λNp, μNq, aNm, bNm, and δNm such that the vector (zN,λN,μN,aN,bN,δN) satisfies the following Karash-Kuhn-Tucker (KKT) condition for problem (3): (40)0=f^N(zN)+𝒥g^N(zN)TλN+𝒥h^N(zN)TμN+𝒥G^N(zN)TaN+𝒥H^N(zN)TbN+𝒥Φ^N(zN)TδN with (41)0λNg^iN(zN)0,Φ^N(z)=H^iN(z)G^iN(z)-tNe,0Φ^N(zN)δN0,0bNH^N(zN)0,0aNG^N(zN)0.

We now prove the almost sure convergence of the regularization SAA method for SMPCC (1).

Theorem 17.

Suppose Assumptions 1011 hold. Let tN0 and let zN be a stationary point of problem (3). If the sequence {zN} converges to z¯ w.p.1 as N and MPCC-MFCQ (Definition 5) holds at z¯, then the following statements hold:

z¯ is a C-stationary point of SMPCC (1) almost surely.

If, in addition, the multipliers ui=0 and vi=0 for all iβJ0, where (42)J0={i{1,2,,m}:G^iN(zN)H^iN(zN)=tN  forinfinitelymanyNG^iN(zN)H^iN}ui=limN(δN)iG^iN(zN),vi=limN(δN)iH^iN(zN),iJ0, then z¯ is a S-stationary point of SMPCC (1) almost surely.

Proof.

Since zN is a stationary point of problem (3), there exist multipliers λNp, μNq, aNm, bNm, and δNm such that (43)0=f^N(zN)+𝒥g^N(zN)TλN+𝒥h^N(zN)TμN+𝒥G^N(zN)TaN+𝒥H^N(zN)TbN+((G^1N(zN)H^1N(zN)+H^1N(zN)G^1N(zN))T(G^1N(zN)H^mN(zN)+H^1N(zN)G^mN(zN))T)TδN with (44)(λN)i0,g^iN(zN)(λN)i=0,i=1,,p,(H^iN(zN)G^iN(zN)-tN)(δN)i=0,i=1,,m,(aN)i0,(bN)i0,H^iN(zN)(bN)i=0,G^iN(zN)(aN)i=0,i=1,2,,m. Then (43) can be reformulated as (45)-f^N(zN)=(𝒥g^N(zN)𝒥h^N(zN)G^iN(zN)T,iJ0cG^iN(zN)T+G^iN(zN)H^iN(zN)H^iN(zN)T,iJ0α¯G^iN(zN)T,iJ0β¯H^iN(zN)T,iJ0cH^iN(zN)T+H^iN(zN)G^iN(zN)G^iN(zN)T,iJ0γ¯H^iN(zN)T,iJ0β¯)TΨN with (46)λNN-p(g^N(zN)),δNN-m(H^N(zN)G^N(zN)-tNe),aNN+m(G^N(zN)),bNN+m(H^N(zN)),viN=(δN)iG^iN(zN),uiN=(δN)iH^iN(zN),i=1,2,,m,ΨN=(λNTμNT(aN)J0cT(uN)J0(α¯β¯)T(bN)J0cT(vN)J0(γ¯β¯)T)T. Next we show that ΨN is almost surely bounded under the MPCC-MFCQ. We assume by contradiction that ΨN is unbounded, then there exists a number sequence {τN}0 such that τNΨNΨ¯0, where (47)Ψ¯=(λ¯Tμ¯Ta¯J0cTu¯J0(α¯β¯)Tb¯J0cTv¯J0(γ¯β¯)T)T. Since (48)H^iN(zN)G^iN(zN)G^iN(zN)T0,iJ0γ¯  w.p.1  as  N,G^iN(zN)H^iN(zN)H^iN(zN)T0,iJ0α¯,w.p.1  as  N and by outer semicontinuousness of normal cone (49)a¯N+m(𝔼[G(z¯,ξ)]),b¯N+m(𝔼[H(z¯,ξ)]),u¯N-m(𝔼[G(z¯,ξ)]𝔼[H(z¯,ξ)]),v¯N-m(𝔼[G(z¯,ξ)]𝔼[H(z¯,ξ)]). Notice that a¯i=0, iJ0cγ¯, and b¯i=0, iJ0cα¯; then by multiplying τN to both sides of (45) and taking limit, we have (50)0=(𝒥𝔼[g(z¯,ξ)]𝒥𝔼[h(z¯,ξ)]𝒥𝔼[G(z¯,ξ)]α¯β¯𝒥𝔼[H(z¯,ξ)]γ¯β¯)TΨ^ with Ψ^0, where (51)Ψ^=(λ¯Tμ¯Ta¯J0c(α¯β¯)Tu¯J0(α¯β¯)Tb¯J0c(γ¯β¯)Tv¯J0(γ¯β¯)T)T. However, we know from MPCC-MFCQ that for any Acmin{𝔼[G(z¯,ξ)] and 𝔼[H(z¯,ξ)]}(52)0int{(𝔼[g(z¯,ξ)]𝔼[h(z¯,ξ)]min{𝔼[G(z¯,ξ)],𝔼[H(z¯,ξ)]})+(𝒥𝔼[g(z¯,ξ)]𝒥𝔼[h(z¯,ξ)]A)n-(-p{0}q{0}m)}, which is called the generalized Robinson constraint qualification in . Notice that for Aicmin{𝔼[Gi(z¯,ξ)] and 𝔼[Hi(z¯,ξ)]}, there exists k[0,1] such that (53)Ai={𝔼[Gi(z¯,ξ)],iα¯𝔼[Hi(z¯,ξ)],iγ¯k𝔼[Gi(z¯,ξ)]+(1-k)𝔼[Hi(z¯,ξ)],iβ¯. Then by dual form of generalized Robinson constraint qualification in Yen , we have for any k[0,1](54)0=(𝒥𝔼[g(z¯,ξ)]𝒥𝔼[h(z¯,ξ)]𝔼[Gi(z¯,ξ)]T,iα¯𝔼[Hi(z¯,ξ)]T,iγ¯k𝔼[Gi(z¯,ξ)]T+(1-k)𝔼[Hi(z¯,ξ)]T,iβ¯)TλλN-p×{0}q×{0}m(0)}λ=0, which means that (55)0=(𝒥𝔼[g(z¯,ξ)]𝒥𝔼[h(z¯,ξ)]𝔼[Gi(z¯,ξ)]T,iα¯𝔼[Hi(z¯,ξ)]T,iγ¯𝔼[Gi(z¯,ξ)]T,iβ¯𝔼[Hi(z¯,ξ)]T,iβ¯)T(μhμgμα¯μγ¯μGμH)μiGμiH0,iβ¯}(μhμgμα¯μγ¯μGμH)=0. That is, Ψ^ in (50) is 0. This contradicts the condition that Ψ^0 and hence {ΨN} is bounded. Without loss of generality, we assume ΨNΨ~ w.p.1 as N, where (56)Ψ~=(λ~Tμ~Ta~J0cTu~J0(α¯β¯)Tb~J0cTv~J0(γ¯β¯)T)T. Notice that (57)(aN)J0ca~J0c,(bN)J0cb~J0cw.p.1,(δN)iH^iN(zN)u~i,iJ0(α¯β¯),(δN)iG^iN(zN)v~i,iJ0(γ¯β¯)  w.p.1. Then we know from (46) that for iβ¯, in the case when iJ0c,a~ib~i0 due to (aN)i0 and (bN)i0 for each N. In the case when iJ0, since (58)(δN)iH^iN(zN)(δN)iG^iN(zN)0, we have u~iv~i0. As a result, by Definition 7, z¯ is a C-stationary point. If Ψ~i=0 for iJ0β¯, then we know from Definition 7 that z¯ is a S-stationary point. The proof is completed.

Remark 18.

For a deterministic MPCC problem, Scholtes  studied the properties of the limit point of a sequence of stationary points generated by the same regularization method under MPCC-LICQ. Notice that MPCC-MFCQ in Theorem 17 is weaker than MPCC-LICQ. Thus this theorem can be seen as an improvement of [17, Theorem 3.1] for solving SMPCC under weaker constraint qualification conditions.

5. Existence and Exponential Convergence Rate

In this section, we discuss the conditions ensuring existence and exponential convergence of stationary points of regularized SAA problem satisfying (40) when the sample size is sufficiently large.

We need the following lemma.

Lemma 19.

Let Xn be a compact set. Suppose Assumptions 1012 hold. Then for any ɛ>0, there exist positive constants C(ɛ) and β(ɛ), independent of N, such that (59)Prob{supzXf^N(z)-𝔼[f(z,ξ(ω))]ɛ}C(ɛ)e-Nβ(ɛ).

Proof.

Under Assumptions 1012, we know from [10, Theorem 7.65] that for each i{1,2,,n}, there exist positive constants Ci(ɛ) and βi(ɛ), independent of N, such that (60)Prob{supzXf^N(z)i-𝔼[f(z,ξ(ω))]iɛn}Ci(ɛ)e-Nβi(ɛ), where f^N(z)i and 𝔼[f(z,ξ(ω))]i denote the ith component of f^N(z) and 𝔼[f(z,ξ(ω))], respectively. Therefore, we have (61)Prob{supzXf^N(z)-𝔼[f(z,ξ(ω))]ɛ}i=1nProb{supzXf^N(z)i-𝔼[f(z,ξ(ω))]iɛn}i=1nCi(ɛ)e-Nβi(ɛ)C(ɛ)e-Nβ(ɛ), where C(ɛ):=nmax{C1(ɛ),C2(ɛ),,Cn(ɛ)}, and β(ɛ):=min{β1(ɛ),β2(ɛ),,βn(ɛ)}.

We now state our existence and exponential convergence results. The proof relies on an application of Robinson’s standard NLP stability theory in .

Theorem 20.

Let z- be a C-stationary point of SMPCC (1) and τN0. Suppose

Assumptions 1012 hold at z-,

MPCC-LICQ (Definition 6), MPCC-SSOSC (Definition 9), and ULSC (Definition 8) hold at z-.

Then we have that

there exits zN satisfying stationary condition (40) of (3) w.p.1 for each N when N is sufficiently large and zNz- w.p.1 as N→∞;

the sequence {zN} in (a) satisfies that for every ɛ>0, there exist positive constants C(ɛ) and β(ɛ), independent of N, such that (62)Prob{zN-z¯ε}C(ε)e-Nβ(ε) for N sufficiently large.

Proof.

Since z- is a C-stationary point of SMPCC, then there exist vectors λ¯|Ig|, μ-q, u¯|α¯|+|β¯|, and v¯|β¯|+|γ¯| such that (63)G(z¯,λ¯,μ¯,u¯,v¯)=0,(64)u¯iv¯i0,iβ¯, where (65)G(z,λ,μ,u,v)=[zL(z,λ,μ,u,v)𝔼[gIg(z,ξ)]𝔼[h(z,ξ)]𝔼[Gα¯β¯(z,ξ)]𝔼[Hβ¯γ¯(z,ξ)]] with (66)L(z,λ,μ,u,v)=𝔼[f(z,ξ)]  +iIgλi𝔼[gi(z,ξ)]+i=1qμi𝔼[hi(z,ξ)]-iα¯β¯ui𝔼[Gi(z,ξ)]-iβ¯γ¯vi𝔼[Hi(z,ξ)]. Notice that (63) can be seen as a KKT condition of the following NLP problem: (67)min𝔼[f(z,ξ)]s.t.𝔼[gi(z,ξ)]  =  0,i    Ig,s.t.𝔼[h(z,ξ)]  =  0,s.t.𝔼[Gi(z,ξ)]  =  0,i  α¯β¯,s.t.𝔼[Hi(z,ξ)]  =  0,i  β¯γ¯  . The MPCC-SSOSC ensures the strong second-order sufficient condition for NLP problem (67), which, under MPCC-LICQ, implies the stability of (67) in the sense of Robinson . Hence, there exist positive numbers ɛ,δ, and c such that for every p𝔹(0,ɛ), the mapping Σ(p)={θn+|Ig|+q+|α¯|+2|β¯|+|γ¯|0G(θ)+p,θ=(z,λ,μ,u,v)} has only one solution θ(p):=(z(p),λ(p),μ(p),u(p),v(p))𝔹(θ¯,δ) with θ¯=(z¯,λ¯,μ¯,u¯,v¯)=z(0) and the mapping θ(·):𝔹(0,ɛ)𝔹(θ¯,δ) satisfying (68)θ(p)-θ(p)cp-pfor  any  p,p'𝔹(0,ɛ).

Since ULSC holds at z¯ and z¯ is a C-stationary point, we have u¯iv¯i>0 for iβ, which means that for sufficiently small δ>0 and any θ𝔹(θ¯,δ), vi/ui  >0, and ui/vi  >0,iβ. Let (69)QN(θ)=[Q1N(θ)𝔼[h(z,ξ)]-h^N(z)𝔼[gIg(z,ξ)]-g^IgN(z)𝔼[Gi(z,ξ)]-G^N(z)+aN,iαβ¯𝔼[Hi(z,ξ)]-H^N(z)+bN,iγ¯β¯], where (70)Q1N(θ)=f^N(z)-𝔼[f(z,ξ)]+iIgλi[𝔼[gi(z,ξ)]-g^iN(z)]+i=1qμi[𝔼[hi(z,ξ)]-h^iN(z)]-iαγ¯ui[𝔼[Gi(z,ξ)]-G^iN(z)]-iβ¯γ¯vi[𝔼[Hi(z,ξ)]-H^iN(z)]+iγuitN(G^iN(z))2G^iN(z)+iαvitN(H^iN(z))2H^iN(z),aN=viuitN,bN=uivitN. For sufficiently small δ>0 and sufficiently large N, H^iN(z)>0, iα, and G^iN(z)>0, iγ. Then by Lemma 13, we have that (71)supθ𝔹(θ-,δ)iγuitN(G^iN(z))2G^iN(z)0,supθ𝔹(θ-,δ)iαvitN(H^iN(z))2H^iN(z)0 w.p.1 as N. By the Uniform Laws of Large Numbers, we have (72)supθ𝔹(θ-,δ)f^N(z)-𝔼[f(z,ξ)]0,supθ𝔹(θ-,δ)λi(g^iN(z)-𝔼[gi(z,ξ)])0,iIg,supθ𝔹(θ-,δ)μi(h^iN(z)-𝔼[hi(z,ξ)])0,i=1,2,,q,supθ𝔹(θ-,δ)ui(G^iN(z)-𝔼[Gi(z,ξ)])0,iα-β-,(73)supθ𝔹(θ-,δ)vi(H^iN(z)-𝔼[Hi(z,ξ)])0,iβ-γ- w.p.1 as N. As a result, combining (70)–(73), we obtain that for ε>0, when N is sufficiently large, (74)Q1Nδ=supθ𝔹(θ-,δ)Q1N(θ)<εw.p.1. In addition, we know from Uniform Laws of Large Numbers that (75)supθ𝔹(θ-,δ)ϱN(z)-𝔼[ϱN(z)]0w.p.1  as  N, which implies that for above ε>0, when N is sufficiently large, (76)supθ𝔹(θ-,δ)ϱN(z)-𝔼[ϱN(z)]<εw.p.1, where ϱN(z) is any element in {g^iN(z), iIg, h^iN(z), i=1,2,,q, H^iN(z), iβ-γ-, G^iN(z), and iβ-α-}. Hence, we know from (69), (74), and (76) that for above ε>0 when N is sufficiently large, (77)QNδ=supθ𝔹(θ-,δ)QN(θ)<εw.p.1. Applying the Brouwer's fixed point theorem to the mapping θ(QN(·)):𝔹(θ-,δ)𝔹(θ-,δ), where θ(·) is defined as in (68), we conclude that there is at least one fixed point θN=(zN,λN,μN,uN,vN)n+|Ig|+q+|α-|+2|β-|+|γ-| such that θN=θ(QN(θN)) w.p.1. Therefore, when N is sufficiently large, there exists θN𝔹(θ-,δ) w.p.1 such that 0G(θN)+QN(θN) w.p.1, namely, (78)0f^N(zN)+iIg(λN)ig^iN(zN)+i=1q(μN)ih^iN(zN)-iα-[(uN)iG^iN(zN)+(uN)itN(HiN(z))2H^iN(zN)]-iγ-[(vN)iH^iN(zN)+(vN)itN(G^iN(zN))2G^iN(zN)]-iβ-[(uN)iG^iN(zN)+(vN)iH^iN(zN)]w.p.1 with (79)h^N(zN)=0,g^IgN(zN)=0,G^iN(zN)H^iN(zN)=tN,iαγ,G^iN(zN)=(vN)i(uN)itN,H^iN(zN)=(uN)i(vN)itN,iβ. Moreover, combining (68) and (77), we obtain (80)θNθ-w.p.1  as  N. Let (81)(δN)i={(uN)iH^iN(zN)iα-,(uN)i(vN)itNiβ-,(vN)iG^iN(zN)iγ-, then we have from (78) that zN is almost surely a stationary point of (3) and (λN,μN,δN) is the corresponding multiplier. Furthermore, by (80), we have zNz-  w.p.1  as  N. The proof of part (a) is completed.

Under condition (ii), we know from (68) and (77) that there exist κ>0 and δ>0 such that (82)zN-z-θN-θ-cQN(zN)cmaxθ𝔹(θ-,δ)QN(θ). For ε>0, combining (70)–(73), we obtain that when N is large enough (83)maxθ𝔹(θ-,δ)Q1N(z)maxz𝔹(z-,δ)f^N(z)-𝔼[f(z,ξ)]+ε4c, which, by (76), means that when N is large enough (84)maxθ𝔹(θ-,δ)QN(z)maxz𝔹(z-,δ)f^N(z)-𝔼[f(z,ξ)]+ε2c. According to Lemma 19, there exist C(ε)>0 and β(ε)>0, independent of N, such that (85)Prob{maxz𝔹(z-,δ)f^N(z)-𝔼[f(z,ξ(ω))]ε}C(ε)e-Nβ(ε), when N is large enough. As a result, the conclusion of (b) follows from (82) and (84).

6. Numerical Results

In this section, we present some preliminary numerical results obtained by the regularization SAA method. Our numerical experiments are carried out in MATLAB 7.1 running on a PC with Intel Pentium M of 1.60 GHz CPU and our tests are focused on different values of the regularization parameter τ and sample size N.

To see the performance of the regularization SAA method, we have also carried out tests for the smoothing SAA method  for (6.3) which incorporates a smoothing NCP scheme based on the following Chen-Harker-Kanzow-Smale (CHKS) smoothing function: (86)ϕτ(a,b)=12(a+b-(a-b)2+4τ2) and compare the test results.

In our experiments, we employed the random number generator unifrnd, exprnd, and normrnd in MATLAB 7.1 to generate independently and identically distributed random samples {ξ1,ξ2,,ξN}. We solved problem (3) with N and τ by the solver fmincon in MATLAB 7.1 to obtain the approximated optimal solution zN. Throughout the tests, we recorded number of iterations of fmincon (Iter) and the values of the objective function at zN (Obj) and these quantities are displayed in the tables of test results.

In the tables below, “REG” and “CHKS” denote regularization SAA method and the smoothing SAA method based on the CHKS smoothing function, respectively.

The examples below varied from examples in Shapiro and Xu .

Example 1.

Consider (87)minf(z)=𝔼[(z1-1)2+(z2-1)2minf(z)=1+(z3-1)2+0.5z42ξ1minf(z)=1+z52+z62ξ3+2ξ2-1]s.t.0𝔼[G(z,ξ)]𝔼[H(z,ξ)]0, where (88)G(z,ξ)=(z4-12z1+ξ3-1,z5-ξ2z2,z6-14ξ1z3)T,H(z,ξ)=(z4-z1+0.25ξ1,z5-z2+ξ2,z6-z3+0.5ξ3)T,ξ=(ξ1,ξ2,ξ3), ξ1,ξ2,ξ3 are independent random variables; ξ1 has an exponential distribution EXP(λ=0.5);ξ2 has an uniform distribution on [0,1]; and ξ3 has a normal distribution 𝒩(μ,σ2) with μ=1 and σ=0.1. The constraint here, which is a complementarity problem, has a solution z=(z1,z2,z3,z4,z5,z6), where (89)zi+3-12zi={12zi-1,if  zi2,0,otherwise, for i=1,2,3. Therefore, substituting above (z3,z4,z5) into the objective function, we obtain that (0.8, 0.8, 0.8, 0.4, 0.4, 0.4) is the exact optimal solution and 0.2 is the optimal value. The test results are presented in Table 1.

The computational results for Example 1.

Methods N                   τ z N Obj Iter
REG 1 0 3 5 × 1 0 - 1 (0.8119, 0.8064, 0.8144, 0.3960, 0.3951, 0.3868) 0.5435 5
CHKS 1 0 3 5 × 1 0 - 1 (0.6503, 0.6444, 0.6542, 0.5947, 0.5868, 0.5784) 1.3664 6

REG 1 0 4 5 × 1 0 - 2 (0.7963, 0.8029, 0.8029, 0.4081, 0.3974, 0.4004) 0.5921 3
CHKS 1 0 4 5 × 1 0 - 2 (0.7937, 0.8004, 0.7996, 0.4109, 0.4004, 0.4030) 0.6021 5

REG 1 0 5 5 × 1 0 - 3 (0.7993, 0.8000, 0.8006, 0.4016, 0.3999, 0.4001) 0.6008 3
CHKS 1 0 5 5 × 1 0 - 3 (0.7993, 0.8001, 0.8004, 0.4017, 0.4000, 0.4000) 0.6009 4
Example 2.

Consider (90)minf(z)=𝔼[2(z1-2)2+2z22minf(z)=1+(z3-3)2ξ2+z42ξ2minf(z)=1+z52ξ1+z62ξ3],s.t.0𝔼[G(z,ξ)]𝔼[H(z,ξ)]0, where G(z,ξ)=(z4-2z1ξ1+ξ2-ξ1,z5+0.5z22ξ2-ξ3,z6-z3ξ3+ξ3)T are H(z,ξ)=(z1,z2,z3), ξ=(ξ1,ξ2,ξ3), ξ1, ξ2, ξ3 are independent random variables; ξ1 has a normal distribution 𝒩(μ,σ2) with μ=0.5 and σ=0.1, ξ2 an exponential distribution EXP(λ=0.5); and ξ3 has a uniform distribution on [0,2]. The constraint has a solution z=(z1,z2,z3,z4,z5,z6), where (91)z4={z1-1.5,if  z11.5,0,otherwise,z5={1-z22,if-1x21,0,otherwise,z6={2z3-2,if  x31,0,otherwise. Therefore, substituting the above (z4,z5,z6) into the objective function, we obtain that (1.75, 0, 0.5, 0.25, 1, 0) is the exact optimal solution and 0.75 is the optimal value. The test results are displayed in Table 2.

Our preliminary numerical results shown in Tables 1 and 2 reveal that our proposed method yields a reasonable solution of the problems considered. To compare with the smoothing SAA method, the regularization SAA method may need fewer iteration numbers.

The computational results for Example 2.

Methods N                   τ z N                         Obj Iter
REG 1 0 3 5 × 1 0 - 1 (1.7521, 0.0002, 0.5000, 0.2485, 0.9800, 0.0001) 0.7257 7
CHKS 1 0 3 5 × 1 0 - 1 (1.6621, 0.0000, 0.4614, 0.5857, 1.1901, 0.3015) 1.7127 10

REG 1 0 4 5 × 1 0 - 2 (1.7502, 0.0001, 0.4999, 0.2498, 0.9990, 0.0000) 0.7485 8
CHKS 1 0 4 5 × 1 0 - 2 (1.7500, 0.0000, 0.5000, 0.2593, 1.0015, 0.0050) 0.7609 11

REG 1 0 5 5 × 1 0 - 3 (1.7498, 0.0000, 0.5000, 0.2504, 0.9999, 0.0000) 0.7503 12
CHKS 1 0 5 5 × 1 0 - 3 (1.7498, 0.0000, 0.5000, 0.2505, 0.9999, 0.0000) 0.7504 11
7. Conclusion

In this paper, we focus on detailed analysis of convergence of a regularization SAA method for SMPCC (1). Almost sure convergence of optimal solutions of the regularized SAA problem is established by the notion of epiconvergence in variational analysis. We improve a convergence result established by Scholtes  on a regularization method for a deterministic MPCC under weaker constraint qualifications. Moreover, the exponential convergence rate of the sequence of Karash-Kuhn-Tucker points generated from the regularized SAA problem is obtained through an application of Robinson's stability theory.

Conflict of Interests

The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence the work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, the paper.

Acknowledgments

The authors are supported by the National Natural Science Foundation of China under Project no. 11201210 and no. 11171138 and Scientific Research Fund of Liaoning Provincial Education Department under Project no. L2012385.

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