Expected Residual Minimization Method for a Class of Stochastic Quasivariational Inequality Problems

Abstract

We consider the expected residual minimization method for a class of stochastic quasivariational inequality problems (SQVIP). The regularized gap function for quasivariational inequality problem (QVIP) is in general not differentiable. We first show that the regularized gap function is differentiable and convex for a class of QVIPs under some suitable conditions. Then, we reformulate SQVIP as a deterministic minimization problem that minimizes the expected residual of the regularized gap function and solve it by sample average approximation (SAA) method. Finally, we investigate the limiting behavior of the optimal solutions and stationary points.

1. Introduction

The quasivariational inequality problem is a very important and powerful tool for the study of generalized equilibrium problems. It has been used to study and formulate generalized Nash equilibrium problem in which a strategy set of each player depends on the other players’ strategies (see, for more details, [1–3]).

QVIP is to find a vector 𝑥^∗∈𝑆(𝑥^∗) such that 𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆𝑥^∗, (1.1) where 𝐹∶ℝ^𝑛→ℝ^𝑛 is a mapping, the symbol ⟨⋅,⋅⟩ denotes the inner product in ℝ^𝑛, and 𝑆∶ℝ^𝑛→2^ℝ𝑛 is a set-valued mapping of which 𝑆(𝑥) is a closed convex set in ℝ^𝑛 for each 𝑥. In particular, if 𝑆 is a closed convex set and 𝑆(𝑥)≡𝑆 for each 𝑥, then QVIP (1.1) becomes the classical variational inequality problem (VIP): find a vector 𝑥^∗∈𝑆 such that 𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆. (1.2)

In most important practical applications, the function 𝐹 always involves some random factors or uncertainties. Let (Ω,ℱ,𝑃) be a probability space. Taking the randomness into account, we get stochastic quasivariational inequality problem (SQVIP): find an 𝑥^∗∈𝑆(𝑥^∗) such that 𝑃𝜔∈Ω∶𝐹𝑥^∗,𝜔,𝑥−𝑥^∗≥0,∀𝑥∈𝑆𝑥^∗=1, (1.3) or equivalently, 𝐹𝑥^∗,𝜔,𝑥−𝑥^∗≥0,∀𝑥∈𝑆𝑥^∗,𝜔∈Ωa.s., (1.4) where 𝐹∶ℝ^𝑛×Ω→ℝ^𝑛 is a mapping and a.s. is abbreviation for “almost surely” under the given probability measure 𝑃.

Due to the introduction of randomness, SQVIP (1.4) becomes more practical and also evokes more and more attentions in the recent literature [4–16]. However, to our best knowledge, most publications in the existing literature discuss the stochastic complementarity problems and the stochastic variational inequality problems, which are two special cases of (1.4). It is well known that quasivariational inequalities are more complicated than variational inequalities and complementarity problems and that they have widely applications. Therefore, it is meaningful and interesting to study the general problem (1.4).

Because of the existence of a random element 𝜔, we cannot generally find a vector 𝑥^∗∈𝑆(𝑥^∗) such that (1.4) holds almost surely. That is, (1.4) is not well defined if we think of solving (1.4) before knowing the realization 𝜔. Therefore, in order to get a reasonable resolution, an appropriate deterministic reformulation for SQVIP becomes an important issue in the study of the considered problem.

Recently, one of the mainstreaming research methods on the stochastic variational inequality problem is expected residual minimization method (see [4, 5, 7, 11–13, 16] and the references therein). Chen and Fukushima [5] formulated the stochastic linear complementarity problem (SLCP) as a minimization problem which minimizes the expectation of gap function (also called residual function) for SLCP. They regarded the optimal solution of this minimization problem as a solution to SLCP. This method is the so-called expected residual minimization method (ERM). Following the ideas of Chen and Fukushima [5], Zhang and Chen [16] considered the stochastic nonlinear complementary problems. Luo and Lin [12, 13] generalized the expected residual minimization method to solve stochastic variational inequality problem.

In this paper, we focus on ERM method for SQVIP. We first show that the regularized gap function for QVIP is differentiable and convex under some suitable conditions. Then, we formulate SQVIP (1.4) as an optimization problem and solve this problem by SAA method.

The rest of this paper is organized as follows. In Section 2, some preliminaries and the reformulation for SQVIP are given. In Section 3, we give some suitable conditions under which the regularized gap function for QVIP is differentiable and convex. In Section 4, we show that the objective function of the reformulation problem is convex and differentiable under some suitable conditions. Finally, the convergence results of optimal solutions and stationary points are given in Section 5.

2. Preliminaries

Throughout this paper, we use the following notations. ‖⋅‖ denotes the Euclidean norm of a vector. For an 𝑛×𝑛 symmetric positive-definite matrix 𝐺, ‖⋅‖_𝐺 denotes the 𝐺-norm defined by ‖𝑥‖_𝐺=⟨𝑥,𝐺𝑥⟩ for 𝑥∈ℝ^𝑛 and Proj_𝑆,𝐺(𝑥) denotes the projection of the point 𝑥 onto the closed convex set 𝑆 with respect to the norm ‖⋅‖_𝐺. For a mapping 𝐹∶ℝ^𝑛→ℝ^𝑛, ∇_𝑥𝐹(𝑥) denotes the usual gradient of 𝐹(𝑥) in 𝑥. It is easy to verify that 𝜆_min‖𝑥‖≤‖𝑥‖_𝐺≤𝜆_max‖𝑥‖, (2.1) where 𝜆_min and 𝜆_max are the smallest and largest eigenvalues of 𝐺, respectively.

The regularized gap function for the QVIP (1.1) is given as follows: 𝑓_𝛼(𝑥)∶=max_{(𝑥)𝑦∈𝑆}−𝐹(𝑥)−𝛼,𝑦−𝑥2𝑦−𝑥²_𝐺, (2.2) where 𝛼 is a positive parameter. Let 𝑋⊆ℝ^𝑛 be defined by 𝑋={𝑥∈ℝ^𝑛∶𝑥∈𝑆(𝑥)}. This is called a feasible set of QVIP (1.1). For the relationship between the regularized gap function (2.2) and QVIP (1.1), the following result has been shown in [17, 18].

Lemma 2.1. Let 𝑓_𝛼(𝑥) be defined by (2.2). Then 𝑓_𝛼(𝑥)≥0 for all 𝑥∈𝑋. Furthermore, 𝑓_𝛼(𝑥^∗)=0 and 𝑥^∗∈𝑋 if and only if 𝑥^∗ is a solution to QVIP (1.1). Hence, problem (1.1) is equivalent to finding a global optimal solution to the problem: min_𝑥∈𝑋𝑓_𝛼(𝑥). (2.3)

Though the regularized gap function 𝑓_𝛼(𝑥) is directional differentiable under some suitable conditions (see, [17, 18]), it is in general nondifferentiable.

The regularized gap function (or residual function) for SQVIP (1.4) is as follows: 𝑓_𝛼(𝑥,𝜔)∶=max_{(𝑥)𝑦∈𝑆}−𝐹()−𝛼𝑥,𝜔,𝑦−𝑥2𝑦−𝑥²_𝐺, (2.4) and the deterministic reformulation for SQVIP is min_𝑥∈𝑋Θ(𝑥)∶=𝔼𝑓_𝛼(𝑥,𝜔), (2.5) where 𝔼 denotes the expectation operator.

Note that the objective function Θ(𝑥) contains mathematical expectation. Throughout this paper, we assume that 𝔼𝑓_𝛼(𝑥,𝜔) cannot be calculated in a closed form so that we will have to approximate it through discretization. One of the most well-known discretization approaches is sample average approximation method. In general, for an integrable function 𝜙∶Ω→ℝ, we approximate the expected value 𝔼[𝜙(𝜔)] with sample average (1/𝑁_𝑘)∑_{𝜔𝑖∈Ω𝑘}𝜙(𝜔_𝑖), where 𝜔₁,…,𝜔_𝑁𝑘 are independently and identically distributed random samples of 𝜔 and Ω_𝑘∶={𝜔₁,…,𝜔_𝑁𝑘}. By the strong law of large numbers, we get the following lemma.

Lemma 2.2. If 𝜙(𝜔) is integrable, then lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜙𝜔_𝑖=𝔼𝜙(𝜔) (2.6) holds with probability one.

Let Θ_𝑘(𝑥)1∶=𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑓_𝛼𝑥,𝜔_𝑖. (2.7) Applying the above techniques, we can get the following approximation of (2.5): min_𝑥∈𝑋Θ_𝑘(𝑥). (2.8)

3. Convexity and Differentiability of 𝑓_𝛼(𝑥)

In the remainder of this paper, we restrict ourself to a special case, where 𝑆(𝑥)=𝑆+𝑚(𝑥). Here, 𝑆 is a closed convex set in ℝ^𝑛 and 𝑚(𝑥)∶ℝ^𝑛→ℝ^𝑛 is a mapping. In this case, we can show that 𝑓_𝛼(𝑥) is continuously differentiable whenever so are the functions 𝐹(𝑥) and 𝑚(𝑥). In order to get this result, we need the following lemma (see [19, Chapter 4, Theorem??1.7]).

Lemma 3.1. Let 𝑆∈ℝ^𝑛 be a nonempty closed set and 𝑈∈ℝ^𝑚 be an open set. Assume that 𝑓∶ℝ^𝑛×𝑈→ℝ be continuous and the gradient ∇_𝑢𝑓(⋅,⋅) is also continuous. If the problem min_𝑥∈𝑆𝑓(𝑥,𝑢) is uniquely attained at 𝑥(𝑢) for any fixed 𝑢∈𝑈, then the function 𝜙(𝑢)∶=min_𝑥∈𝑆𝑓(𝑥,𝑢) is continuously differentiable and ∇_𝑢𝜙(𝑢) is given by ∇_𝑢𝜙(𝑢)=∇_𝑢𝑓(𝑥(𝑢),𝑢).

For any 𝑦∈𝑆(𝑥)=𝑆+𝑚(𝑥), we can find a vector 𝑧∈𝑆 such that 𝑦=𝑧+𝑚(𝑥). Thus, we can rewrite (2.2) as follows: 𝑓_𝛼(𝑥)=max_𝑧∈𝑆−⟨𝐹(𝑥)(𝑥)⟩−𝛼,𝑧+𝑚−𝑥2‖𝑧−(𝑥−𝑚(𝑥))‖²_𝐺=−min_𝑧∈𝑆𝐹(𝑥)((𝑥+𝛼⟨,𝑧−𝑥−𝑚))⟩2‖𝑧−(𝑥−𝑚(𝑥))‖²_𝐺. (3.1) The minimization problem in (3.1) is essentially equivalent to the following problem: min_𝑧∈𝑆𝑧−𝑥−𝑚(𝑥)−𝛼⁻¹𝐺⁻¹𝐹(𝑥)²_𝐺. (3.2)

It is easy to know that problem (3.2) has a unique optimal solution Proj_𝑆,𝐺(𝑥−𝑚(𝑥)−𝛼⁻¹𝐺⁻¹𝐹(𝑥)). Thus, Proj_𝑆,𝐺(𝑥−𝑚(𝑥)−𝛼⁻¹𝐺⁻¹𝐹(𝑥)) is also a unique solution of problem (3.1). The following result is a natural extension of [20, Theorem??3.2].

Theorem 3.2. If 𝑆 is a closed convex set in ℝ^𝑛 and 𝑚(𝑥) and 𝐹(𝑥) are continuously differentiable, then the regularized gap function 𝑓_𝛼(𝑥) given by (2.2) is also continuously differentiable and its gradient is given by ∇𝑓_𝛼(𝑥)=[𝐼−∇𝑚(𝑥)]𝐹(𝑥)−[∇𝐹(𝑥)−𝛼(𝐼−∇𝑚(𝑥))𝐺]𝑧_𝛼(𝑥)−(𝑥−𝑚(𝑥)), (3.3) where 𝑧_𝛼(𝑥)=𝑃𝑟𝑜𝑗_𝑆,𝐺(𝑥−𝑚(𝑥)−𝛼⁻¹𝐺⁻¹𝐹(𝑥)) and 𝐼 denotes the 𝑛×𝑛 identity matrix.

Proof. Let us define the function ℎ∶ℝ^𝑛×𝑆→ℝ by ℎ()=⟨𝐹(𝑥)((𝑥+𝛼𝑥,𝑧,𝑧−𝑥−𝑚))⟩2‖𝑧−(𝑥−𝑚(𝑥))‖²_𝐺. (3.4) It is obviously that if 𝐹(𝑥) and 𝑚(𝑥) are continuous, then ℎ(𝑥,𝑧) is continuous in (𝑥,𝑧). If 𝐹(𝑥) and 𝑚(𝑥) are continuously differentiable, then ∇_𝑥ℎ(𝑥,𝑧)=−[𝐼−∇𝑚(𝑥)]𝐹(𝑥)+[∇𝐹(𝑥)−𝛼(𝐼−∇𝑚(𝑥))𝐺][𝑧−(𝑥−𝑚(𝑥))] (3.5) is continuous in (𝑥,𝑧). By (3.1), we have 𝑓_𝛼(𝑥)=−min_𝑧∈𝑆ℎ(𝑥,𝑧). (3.6) Since the minimum on the right-hand side of (3.6) is uniquely attained at 𝑧=𝑧_𝛼(𝑥), it follows from Lemma 3.1 that 𝑓_𝛼(𝑥) is differentiable and its gradient is given by ∇𝑓_𝛼(𝑥)=−∇_𝑥ℎ𝑥,𝑧_𝛼(𝑥)=[𝐼−∇𝑚(𝑥)]𝐹(𝑥)−[∇𝐹(𝑥)−𝛼(𝐼−∇𝑚(𝑥))𝐺]𝑧_𝛼(𝑥)−(𝑥−𝑚(𝑥)). (3.7) This completes the proof.

Remark 3.3. When 𝑚(𝑥)≡0, we have 𝑆(𝑥)≡𝑆 and so QVIP (1.1) reduces to VIP (1.2). In this case ∇𝑓_𝛼(𝑥)=𝐹(𝑥)−[∇𝐹(𝑥)−𝛼𝐺]𝑧_𝛼(𝑥)−𝑥, (3.8) where 𝑧_𝛼(𝑥)=Proj_𝑆,𝐺𝑥−𝛼⁻¹𝐺⁻¹𝐹(𝑥). (3.9) Moreover, when 𝛼=1, we have ∇𝑓_𝛼(𝑥)=𝐹(𝑥)−[∇𝐹(𝑥)−𝐺]𝑧_𝛼(𝑥),𝑧−𝑥_𝛼(𝑥)=Proj_𝑆,𝐺𝑥−𝐺⁻¹𝐹(𝑥), (3.10) which is the same as [20, Theorem??3.2].

Now we investigate the conditions under which 𝑓_𝛼(𝑥) is convex.

Theorem 3.4. Suppose that 𝐹(𝑥)=𝑀𝑥+𝑞 and 𝑚(𝑥)=𝑁𝑥, where 𝑀 and 𝑁 are 𝑛×𝑛 matrices and 𝑞∈ℝ^𝑛 is a vector. Denote 𝛽_min and 𝜇_max by the smallest and largest eigenvalues of 𝑀^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀 and (𝑁−𝐼)^𝑇𝐺(𝑁−𝐼), respectively. We have the following statements.(i) If 𝜇_max>0, 𝛽_min≥0 and 𝛼≤(𝛽_min/𝜇_max), then the function 𝑓_𝛼(𝑥) is convex. Moreover, if there exists a constant 𝛽>0 such that 𝛼≤(𝛽_min/𝜇_max(1+𝛽)), then 𝑓_𝛼(𝑥) is strongly convex with modulus 𝛼𝛽𝜇_max. (ii)If 𝜇_max=0 and 𝛽_min≥0, then the function 𝑓_𝛼(𝑥) is convex. Moreover, if 𝛽_min>0, then 𝑓_𝛼(𝑥) is strongly convex with modulus 𝛽_min.

Proof. Substituting 𝐹(𝑥)=𝑀𝑥+𝑞 and 𝑚(𝑥)=𝑁𝑥 into (3.1), we have 𝑓_𝛼(𝑥)=max_𝑧∈𝑆−()𝑥−𝛼𝑀𝑥+𝑞,𝑧+𝑁−𝐼2‖𝑧−(𝐼−𝑁)𝑥‖²_𝐺. (3.11) Define 𝐻()()𝑥−𝛼𝑥,𝑧=−𝑀𝑥+𝑞,𝑧+𝑁−𝐼2‖𝑧−(𝐼−𝑁)𝑥‖²_𝐺. (3.12) Noting that ∇²_𝑥𝐻(𝑥,𝑧)=𝑀^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀−𝛼(𝑁−𝐼)^𝑇𝐺(𝑁−𝐼), (3.13) we have, for any 𝑦∈ℝ^𝑛, 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧)𝑦=𝑦^𝑇𝑀^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀𝑦−𝛼𝑦^𝑇(𝑁−𝐼)^𝑇𝐺()𝑦≥𝛽𝑁−𝐼_min−𝛼𝜇_max𝑦². (3.14)
If 𝜇_max>0, 𝛽_min≥0 and 𝛼≤(𝛽_min/𝜇_max), we have 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧)𝑦≥𝛽_min−𝛼𝜇_max𝑦²≥0. (3.15) This implies that the Hessen matrix ∇²_𝑥𝐻(𝑥,𝑧) is positive semidefinite and hence 𝐻(𝑥,𝑧) is convex in 𝑥 for any 𝑧∈𝑆. In consequence, by (3.11), the regularized gap function 𝑓_𝛼(𝑥) is convex. Moreover, if 𝛼≤(𝛽_min/𝜇_max(1+𝛽)), then 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧)𝑦≥𝛽_min−𝛼𝜇_max𝑦²≥𝛼𝛽𝜇_max𝑦², (3.16) which means that 𝐻(𝑥,𝑧) is strongly convex with modulus 𝛼𝛽𝜇_max in 𝑥 for any 𝑧∈𝑆. From (3.11), we know that the regularized gap function 𝑓_𝛼(𝑥) is strongly convex.
If 𝜇_max=0 and 𝛽_min≥0, we have 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧)𝑦≥𝛽_min𝑦²≥0. (3.17) Thus, the regularized gap function 𝑓_𝛼(𝑥) is convex. Moreover, if 𝛽_min>0, then the regularized gap function 𝑓_𝛼(𝑥) is strongly convex with modulus 𝛽_min. This completes the proof.

Remark 3.5. When 𝑁=0, QVIP (1.1) reduces to VIP (1.2). Denote 𝛽_min and 𝜇_max by the smallest and largest eigenvalues of 𝑀^𝑇+𝑀 and 𝐺, respectively. In this case, the function 𝑓_𝛼(𝑥)=max_𝑧∈𝑆−⟨𝐹(𝑥)⟩−𝛼,𝑧−𝑥2‖𝑧−𝑥‖²_𝐺 (3.18) is convex when 𝜇_max>0, 𝛽_min≥0 and 𝛼≤(𝛽_min/𝜇_max).

Remark 3.6. When 𝑁=0 and 𝐺=𝐼, we have that 𝜇_max=1. In this case, the function 𝑓_𝛼(𝑥)=max_𝑧∈𝑆−⟨𝐹(𝑥)⟩−𝛼,𝑧−𝑥2‖𝑧−𝑥‖² (3.19) is convex when 𝛽_min≥0 and 𝛼≤𝛽_min. This is consistent with [4, Theorem??2.1].

4. Properties of Function Θ

In this section, we consider the properties of the objective function Θ(𝑥) of problem (2.5). In what follows we show that Θ(𝑥) is differentiable under some suitable conditions.

Theorem 4.1. Suppose that 𝐹(𝑥,𝜔)∶=𝑀(𝜔)𝑥+𝑄(𝜔), where 𝑀∶Ω→ℝ^𝑛×𝑛 and 𝑄∶Ω→ℝ^𝑛 with 𝔼‖𝑀(𝜔)‖²+‖𝑄(𝜔)‖²<+∞. (4.1) Let 𝑆(𝑥)=𝑆+𝑁𝑥. Then the function Θ(𝑥) is differentiable and ∇_𝑥Θ(𝑥)=𝔼∇_𝑥𝑓_𝛼(𝑥,𝜔). (4.2)

Proof. Since 𝑆(𝑥)=𝑆+𝑁𝑥, it is easy to know that 𝑓_𝛼(𝑥,𝜔)=−𝐹(𝑥,𝜔),𝑦_𝛼()−(−𝛼𝑥,𝜔𝑥−𝑁𝑥)2𝑦_𝛼(𝑥,𝜔)−(𝑥−𝑁𝑥)²_𝐺, (4.3) where 𝑦_𝛼(𝑥,𝜔)=Proj_𝑆,𝐺𝑥−𝑁𝑥−𝛼⁻¹𝐺⁻¹𝐹(𝑥,𝜔). (4.4) It follows from Lemma 2.1 that 𝑓_𝛼(𝑥,𝜔)≥0 and so 𝛼2𝑦_𝛼(𝑥,𝜔)−𝑥+𝑁𝑥²_𝐺≤−𝐹(𝑥,𝜔),𝑦_𝛼()≤‖𝐹(𝑦𝑥,𝜔−𝑥+𝑁𝑥𝑥,𝜔)‖_𝛼()≤1𝑥,𝜔−𝑥+𝑁𝑥𝜆_min‖𝐹(𝑥,𝜔)‖𝑦_𝛼(𝑥,𝜔)−𝑥+𝑁𝑥_𝐺. (4.5) Thus, 𝑦_𝛼(𝑥,𝜔)−𝑥+𝑁𝑥_𝐺≤2𝛼𝜆_min‖𝐹(,𝑦𝑥,𝜔)‖_𝛼()≤1𝑥,𝜔−𝑥+𝑁𝑥𝜆_min𝑦_𝛼(𝑥,𝜔)−𝑥+𝑁𝑥_𝐺≤2𝛼𝜆_min‖𝐹(𝑥,𝜔)‖. (4.6)
In a similar way to Theorem 3.2, we can show that 𝑓_𝛼(𝑥,𝜔) is differentiable with respect to 𝑥 and ∇_𝑥𝑓_𝛼(𝑥,𝜔)=(𝐼−𝑁)𝐹(𝑥,𝜔)−[𝑀(𝜔)−𝛼(𝐼−𝑁)𝐺]𝑦_𝛼(𝑥,𝜔)−(𝐼−𝑁)𝑥. (4.7) It follows that ∇_𝑥𝑓_𝛼(𝑥,𝜔)≤‖𝐼−𝑁‖‖𝐹(𝑥,𝜔)‖−‖𝑀(𝜔)−𝛼(𝐼−𝑁)𝐺‖𝑦_𝛼()−()𝑥≤‖+2𝑥,𝜔𝐼−𝑁‖𝐼−𝑁𝛼𝜆_min‖𝑀(𝜔)()𝐺𝐹(≤2‖𝐺‖−𝛼𝐼−𝑁‖‖𝑥,𝜔)‖1+𝜆_min‖𝐹(+2𝐼−𝑁‖‖𝑥,𝜔)‖𝛼𝜆_min‖𝑀(𝜔𝐹(≤2‖𝐺‖)‖‖𝑥,𝜔)‖1+𝜆_min‖‖𝑥𝑀(𝜔+‖𝑄(𝜔+2𝐼−𝑁‖(1+‖)(‖)‖)‖)𝛼𝜆_min(1+‖𝑥‖)(‖𝑀(𝜔)‖+‖𝑄(𝜔)‖)²≤2‖𝐺‖1+𝜆_min‖‖𝑥𝑀(𝜔+‖𝑄(𝜔+4𝐼−𝑁‖(1+‖)(‖)‖)‖)𝛼𝜆_min(1+‖𝑥‖)‖𝑀(𝜔)‖²+‖𝑄(𝜔)‖². (4.8) By [21, Theorem??16.8], the function Θ is differentiable and ∇_𝑥Θ(𝑥)=𝔼∇_𝑥𝑓_𝛼(𝑥,𝜔). This completes the proof.

The following theorem gives some conditions under which Θ(𝑥) is convex.

Theorem 4.2. Suppose that the assumption of Theorem 4.1 holds. Let 𝛽₀∶=inf_{𝜔∈Ω⧵Ω0}𝜆_min𝑀(𝜔)^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀(𝜔), (4.9) where Ω₀ is a null subset of Ω and 𝜆_min(𝐺) denotes the smallest eigenvalue of 𝐺. We have the following statements. (i) If 𝜇_max>0, 𝛽₀>0 and 𝛼≤(𝛽₀/𝜇_max), then the function Θ(𝑥) is convex. Moreover, if 𝛼≤(𝛽₀/𝜇_max(1+𝛽)) with 𝛽>0, then Θ(𝑥) is strongly convex with modulus 𝛼𝛽𝜇_max. (ii)If 𝜇_max=0 and 𝛽₀≥0, then the function Θ(𝑥) is convex. Moreover, if 𝛽₀>0, then Θ𝛼(𝑥) is strongly convex with modulus 𝛽₀.

Proof. Define 𝐻()⟨𝑀(𝜔)(𝜔)()𝑥⟩−𝛼𝑥,𝑧,𝜔=−𝑥+𝑄,𝑧+𝑁−𝐼2‖𝑧−(𝐼−𝑁)𝑥‖²_𝐺. (4.10) Noting that ∇²_𝑥𝐻(𝑥,𝑧,𝜔)=𝑀(𝜔)^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀(𝜔)−𝛼(𝑁−𝐼)^𝑇𝐺(𝑁−𝐼), (4.11) we have, for any 𝑦∈ℝ^𝑛, 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧,𝜔)𝑦=𝑦^𝑇𝑀(𝜔)^𝑇(𝐼−𝑁)+(𝐼−𝑁)^𝑇𝑀(𝜔)𝑦−𝛼𝑦^𝑇(𝑁−𝐼)^𝑇𝐺()𝑦≥𝛽𝑁−𝐼₀−𝛼𝜇_max𝑦², (4.12) where the inequality holds almost surely.
If 𝜇_max>0, 𝛽₀>0 and 𝛼≤(𝛽₀/𝜇_max), then 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧,𝜔)𝑦≥0. (4.13) This implies that the Hessen matrix ∇²_𝑥𝐻(𝑥,𝑧,𝜔) is positive semidefinite and hence 𝐻(𝑥,𝑧,𝜔) is convex in 𝑥 for any 𝑧∈𝑆. Since 𝑓_𝛼(𝑥,𝜔)=max_{(𝑥)𝑦∈𝑆}−𝐹()−𝛼𝑥,𝜔,𝑦−𝑥2𝑦−𝑥²_𝐺=max_𝑧∈𝑆𝐻(𝑥,𝑧,𝜔), (4.14) the regularized gap function 𝑓_𝛼(𝑥,𝜔) is convex and so is Θ(𝑥). Moreover, if 𝛼≤(𝛽₀/𝜇_max(1+𝛽)), then 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧,𝜔)𝑦≥𝛼𝛽𝜇_max𝑦², (4.15) which means that 𝐻(𝑥,𝑧,𝜔) is strongly convex in 𝑥 for any 𝑧∈𝑆. From the definitions of 𝐻(𝑥,𝑧,𝜔) and 𝑓_𝛼(𝑥,𝜔), we know that 𝑓_𝛼(𝑥,𝜔) is strongly convex with modulus 𝛼𝛽𝜇_max and so is Θ(𝑥).
If 𝜇_max=0 and 𝛽₀≥0, then 𝑦^𝑇∇²_𝑥𝐻(𝑥,𝑧,𝜔)𝑦≥𝛽₀𝑦²≥0, (4.16) which implies that the regularized gap function 𝑓_𝛼(𝑥,𝜔) is convex and so is Θ(𝑥). Moreover, if 𝛽₀>0, then Θ(𝑥) is strongly convex with modulus 𝛽₀. This completes the proof.

It is easy to verify that 𝑋={𝑥∈ℝ^𝑛∶𝑥∈𝑆(𝑥)} is a convex subset when 𝑆(𝑥)=𝑆+𝑁𝑥. Thus, Theorem 4.2 indicates that problem (2.5) is a convex program. From the proof details of Theorem 4.2, we can also get that problem (2.8) is a convex program. Hence we can obtain a global optimal solution using existing solution methods.

5. Convergence of Solutions and Stationary Points

In this section, we will investigate the limiting behavior of the optimal solutions and stationary points of (2.8).

Note that if the conditions of Theorem 4.1 are satisfied, then the set 𝑋 is closed, and 𝔼‖𝑀(𝜔)‖<∞,𝔼[‖𝑀(𝜔)‖+‖𝑄(𝜔)‖+𝑐]²<∞, (5.1) where 𝑐 is a constant.

Theorem 5.1. Suppose that the conditions of Theorem 4.1 are satisfied. Let 𝑥^𝑘 be an optimal solution of problem (2.8) for each 𝑘. If 𝑥^∗ is an accumulation point of {𝑥^𝑘}, then it is an optimal solution of problem (2.5).

Proof. Without loss of generality, we assume that 𝑥^𝑘 itself converges to 𝑥^∗ as 𝑘 tends to infinity. It is obvious that 𝑥^∗∈𝑋.
We first show that lim_𝑘→∞Θ_𝑘𝑥^𝑘−Θ_𝑘𝑥^∗=0. (5.2) It follows from mean-value theorem that Θ_𝑘𝑥^𝑘−Θ_𝑘𝑥^∗=1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑓_𝛼𝑥^𝑘,𝜔_𝑖−𝑓_𝛼𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑓_𝛼𝑥^𝑘,𝜔_𝑖−𝑓_𝛼𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}∇_𝑥𝑓_𝛼𝑦^𝑘_𝑖,𝜔_𝑖𝑥^𝑘−𝑥^∗, (5.3) where 𝑦^𝑘_𝑖=𝛾^𝑘_𝑖𝑥^𝑘+(1−𝛾^𝑘_𝑖)𝑥^∗ and 𝛾^𝑘_𝑖∈[0,1]. From the proof details of Theorem 4.1, we have ∇_𝑥𝑓_𝛼𝑦^𝑘_𝑖,𝜔_𝑖≤2‖𝐺‖1+𝜆_min1+𝑦^𝑘_𝑖𝑀𝜔_𝑖+𝑄𝜔_𝑖‖‖+2𝐼−𝑁𝛼𝜆_min1+𝑦^𝑘_𝑖𝑀𝜔_𝑖+𝑄𝜔_𝑖². (5.4) Since lim_𝑘→+∞𝑥^𝑘=𝑥^∗, there exists a constant 𝐶 such that ‖𝑥^𝑘‖≤𝐶 for each 𝑘. By the definition of 𝑦^𝑘_𝑖, we know that ‖𝑦^𝑘_𝑖‖≤𝐶. Hence, ∇_𝑥𝑓_𝛼𝑦^𝑘_𝑖,𝜔_𝑖≤2‖𝐺‖1+𝜆_min(1+𝐶)𝑀𝜔_𝑖+𝑄𝜔_𝑖‖‖+2𝐼−𝑁𝛼𝜆_min(1+𝐶)𝑀𝜔_𝑖+𝑄𝜔_𝑖²≤𝐶^′𝑀𝜔_𝑖+𝑄𝜔_𝑖+1², (5.5) where 𝐶^′2‖𝐺‖=max1+𝜆_min(‖,21+𝐶)‖𝐼−𝑁𝛼𝜆_min(1+𝐶). (5.6) It follows that Θ_𝑘𝑥^𝑘−Θ_𝑘𝑥^∗≤𝐶^′1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖+𝑄𝜔_𝑖+1²𝑥^𝑘−𝑥^∗→0, (5.7) which means that (5.2) holds.
Now, we show that 𝑥^∗ is an optimal solution of problem (2.5). It follows from (5.2) and Θ_𝑘𝑥^𝑘−Θ𝑥^∗≤Θ_𝑘𝑥^𝑘−Θ_𝑘𝑥^∗+Θ_𝑘𝑥^∗−Θ𝑥^∗, (5.8) that lim_𝑘→+∞Θ_𝑘(𝑥^𝑘)=Θ(𝑥^∗). Since 𝑥^𝑘 is an optimal solution of problem (2.8) for each 𝑘, we have that, for any 𝑥∈𝑋, Θ_𝑘𝑥^𝑘≤Θ_𝑘(𝑥). (5.9) Letting 𝑘→∞ above, we get from (5.2) and Lemma 2.2 that Θ𝑥^∗≤Θ(𝑥), (5.10) which means 𝑥^∗ is an optimal solution of problem (2.5). This completes the proof.

In general, it is difficult to obtain a global optimal solution of problem (2.8), whereas computation of stationary points is relatively easy. Therefore, it is important to study the limiting behavior of stationary points of problem (2.8).

Definition 5.2. 𝑥^𝑘 is said to be stationary to problem (2.8) if ∇_𝑥Θ_𝑘𝑥^𝑘,𝑦−𝑥^𝑘≥0,∀𝑦∈𝑋, (5.11) and 𝑥^∗ is said to be stationary to problem (2.5) if ∇_𝑥Θ𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝑋. (5.12)

Theorem 5.3. Let 𝑥^𝑘 be stationary to problem (2.8) for each 𝑘. If the conditions of Theorem 4.1 are satisfied, then any accumulation point 𝑥^∗ of {𝑥^𝑘} is a stationary point of problem (2.5).

Proof. Without loss of generality, we assume that {𝑥^𝑘} itself converges to 𝑥^∗.
At first, we show that lim_𝑘→∞∇_𝑥Θ_𝑘𝑥^𝑘−∇_𝑥Θ_𝑘𝑥^∗=0. (5.13) It follows from (2.1) and the nonexpansivity of the projection operator that 𝑦_𝛼𝑥^𝑘,𝜔−𝑦_𝛼𝑥^∗≤1,𝜔𝜆_min𝑦_𝛼(𝑥^𝑘,𝜔)−𝑦_𝛼(𝑥^∗,𝜔)_𝐺=1𝜆_minProj_𝑆,𝐺𝑥^𝑘−𝑁𝑥^𝑘−𝛼⁻¹𝐺⁻¹𝐹𝑥^𝑘,𝜔−Proj_𝑆,𝐺𝑥^∗−𝑁𝑥^∗−𝛼⁻¹𝐺⁻¹𝐹𝑥^∗,𝜔_𝐺≤1𝜆_min𝑥^𝑘−𝑁𝑥^𝑘−𝛼⁻¹𝐺⁻¹𝐹𝑥^𝑘,𝜔−𝑥^∗−𝑁𝑥^∗−𝛼⁻¹𝐺⁻¹𝐹𝑥^∗,𝜔_𝐺≤𝜆_max𝜆_min𝑥^𝑘−𝑁𝑥^𝑘−𝛼⁻¹𝐺⁻¹𝐹𝑥^𝑘,𝜔−𝑥^∗−𝑁𝑥^∗−𝛼⁻¹𝐺⁻¹𝐹𝑥^∗≤,𝜔𝜆_max𝜆_min‖𝐼−𝑁‖+𝛼⁻¹𝐺⁻¹‖𝑀(𝜔)‖𝑥^𝑘−𝑥^∗. (5.14) Thus, ∇_𝑥Θ_𝑘𝑥^𝑘−∇_𝑥Θ_𝑘𝑥^∗≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}∇_𝑥𝑓_𝛼𝑥^𝑘,𝜔_𝑖−∇_𝑥𝑓_𝛼𝑥^∗,𝜔_𝑖=1𝑁_{𝑘𝜔𝑖∈Ω𝑘}(𝐼−𝑁)𝑀𝜔_𝑖𝑥^𝑘+𝑄𝜔_𝑖−𝑀𝜔_𝑖()𝐺×𝑦−𝛼𝐼−𝑁_𝛼𝑥^𝑘,𝜔_𝑖−(𝐼−𝑁)𝑥^𝑘−(𝐼−𝑁)𝑀𝜔_𝑖𝑥^∗+𝑄𝜔_𝑖−𝑀𝜔_𝑖()𝐺×𝑦−𝛼𝐼−𝑁_𝛼𝑥^∗,𝜔_𝑖−(𝐼−𝑁)𝑥^∗≤2𝑥^𝑘−𝑥^∗‖1‖𝐼−𝑁𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖+𝛼‖𝐼−𝑁‖²‖𝐺‖𝑥^𝑘−𝑥^∗+1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖−𝛼(𝐼−𝑁)𝐺𝑦_𝛼𝑥^𝑘,𝜔_𝑖−𝑦_𝛼𝑥^∗,𝜔_𝑖≤2𝑥^𝑘−𝑥^∗‖1‖𝐼−𝑁𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖+𝛼‖𝐼−𝑁‖²‖𝐺‖𝑥^𝑘−𝑥^∗+𝜆_max𝜆_min1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖−𝛼(𝐼−𝑁)𝐺‖𝐼−𝑁‖+𝛼⁻¹𝐺⁻¹𝑀𝜔_𝑖𝑥^𝑘−𝑥^∗≤2+𝜆_max𝜆_min‖𝐺‖𝐺⁻¹‖1+1‖𝐼−𝑁𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖𝑥^𝑘−𝑥^∗+1+𝜆_max𝜆_min𝛼‖𝐼−𝑁‖²‖𝐺‖𝑥^𝑘−𝑥^∗+𝛼⁻¹𝐺⁻¹𝜆_max𝜆_min1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝑀𝜔_𝑖²𝑥^𝑘−𝑥^∗→0, (5.15) which means that (5.13) is true.
Next, we show that lim_𝑘→∞∇_𝑥Θ_𝑘𝑥^𝑘=∇_𝑥Θ𝑥^∗. (5.16) It follows from Lemma 2.2 and Theorem 4.1 that lim_𝑘→∞∇_𝑥Θ_𝑘𝑥^∗=lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}∇_𝑥𝑓_𝛼𝑥^∗,𝜔_𝑖=𝔼∇_𝑥𝑓_𝛼𝑥^∗,𝜔=∇_𝑥Θ𝑥^∗. (5.17) By (5.13), we have ∇_𝑥Θ_𝑘𝑥^𝑘−∇_𝑥Θ𝑥^∗≤∇_𝑥Θ_𝑘𝑥^𝑘−∇_𝑥Θ_𝑘𝑥^∗+∇_𝑥Θ_𝑘𝑥^∗−∇_𝑥Θ𝑥^∗→0, (5.18) which implies that (5.16) is true.
Now we show that 𝑥^∗ is a stationary point of problem (2.5). Since 𝑥^𝑘 is stationary to problem (2.8), that is, for any 𝑦∈𝑋, ∇_𝑥Θ_𝑘𝑥^𝑘,𝑦−𝑥^𝑘≥0. (5.19) Letting 𝑘→∞ above, we get from (5.16) that ∇_𝑥Θ𝑥^∗,𝑦−𝑥^∗≥0. (5.20) Thus, 𝑥^∗ is a stationary point of problem (2.5). This completes the proof.

Acknowledgments

This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (111171237, 71101099).