Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Abstract

Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochastic 𝑃-function, stochastic 𝑃₀-function, and stochastic uniformly 𝑃-function. Furthermore, the conditions such that the function is a stochastic 𝑃(𝑃₀)-function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.

1. Introduction

Given a vector-valued function 𝐹∶𝐑^𝑛×Ω→𝐑^𝑛, the stochastic complementarity problems, denoted by SCP(𝐹(𝑥,𝜔)), are to find a vector 𝑥^∗ such that 𝑥^∗≥0,𝐹𝑥^∗𝑥,𝜔≥0,_∗𝑇𝐹𝑥^∗,𝜔=0,𝜔∈Ωa.s., (1) where 𝜔∈Ω⊆𝐑^𝑚 is a random vector with given probability distribution 𝒫 and “a.s." means “almost surely" under the given probability measure. Particularly, when 𝐹 is an affine function of 𝑥 for any 𝜔, that is, 𝐹(𝑥,𝜔)=𝑀(𝜔)𝑥+𝑞(𝜔),𝜔∈Ω, (2) where 𝑀(𝜔)∈𝐑^𝑛×𝑛 and 𝑞(𝜔)∈𝐑^𝑛, the SCP(𝐹(𝑥,𝜔)) is called stochastic linear complementarity problems, denoted by SLCP(𝑀(𝜔),𝑞(𝜔)). Correspondingly, problem (1) is called stochastic nonlinear complementarity problem, denoted by SNCP(𝐹(𝑥,𝜔)) if 𝐹 can not be denoted by an affine function of 𝑥 for any 𝜔. The deterministic problems, which are called complementarity problems (denoted by CP(𝐹(𝑥))), have been intensively studied. More information about theoretical analysis, numerical algorithms and applications especially in economics and engineering can be found in comprehensive books [1, 2].

In practical applications, some elements may involve stochastic factors. In fact, due to stochastic factors, the function value of 𝐹 depends not only on 𝑥, but also on random vectors. Hence, problem (1) does not have solution in general for almost all 𝜔∈Ω. To solve these problems, researchers focus on giving reasonable deterministic reformulations for SCP(𝐹(𝑥,𝜔)). Certainly, different deterministic formulations may yield different solutions that are optimal in different senses. In the study of SCP(𝐹(𝑥,𝜔)), three types of formulations have been proposed; the expected value (EV) formulation, the expected residual minimization (ERM) formulation, and the SMPEC formulation [3].

The EV formulation is studied by Gürkan et al. [4]. The problem considered in [4] is actually a stochastic variational inequality problem. When applied to the SCP(𝐹(𝑥,𝜔)), the EV model can be stated as follows: 𝑥^∗≥0,𝐄𝐹𝑥^∗,𝜔≥0,𝑥^∗𝑇𝐄𝐹𝑥^∗,𝜔=0, (3) where 𝐄 means expectation with respect to 𝜔.

The ERM model is first proposed by Chen and Fukushima [5] for solving the SLCP(𝑀(𝜔),𝑞(𝜔)). By employing an NCP function 𝜙, the SCP(𝐹(𝑥,𝜔)) (1) is transformed equivalently to the stochastic equations Φ(𝑥,𝜔)=0,𝜔∈Ωa.s., (4) where Φ∶𝐑^𝑛×Ω→𝐑^𝑛 is defined by Φ()𝜙𝑥𝑥,𝜔∶=¹,𝐹₁()⋮𝜙𝑥𝑥,𝜔_𝑛,𝐹_𝑛(𝑥,𝜔), (5) and 𝑥_𝑖 denotes the 𝑖th component of the vector 𝑥. Here 𝜙∶𝐑^𝑛→𝐑 is an NCP function which has the property 𝜙(𝑎,𝑏)=0⟺𝑎≥0,𝑏≥0,𝑎𝑏=0. (6) Then the ERM formulation for (1) is given by min_{𝑥∈𝐑𝑛+}𝜃(𝑥)∶=𝐄‖Φ(𝑥,𝜔)‖². (7) The NCP functions employed in [5] include the Fischer-Burmeister function, which is defined by 𝜙_FB(𝑎,𝑏)∶=𝑎²+𝑏²−(𝑎+𝑏) (8) and the min function 𝜙_min(𝑎,𝑏)∶=min{𝑎,𝑏}. (9) In particular, it is known [6, 7] that there exist the following relations between these two functions: 2√𝜙2+2^min≤𝜙_FB≤√2+2𝜙_min. (10) As observed in [5], the ERM formulations with different NCP functions may have different properties. Subsequently, the ERM formulation for SCP(𝐹(𝑥,𝜔)) has been studied in [6, 8–13]. Note that Fang et al. [8] propose a new concept of stochastic matrice: 𝑀(⋅) is called a stochastic 𝑅₀ matrix if 𝒫𝜔∶𝑥≥0,𝑀(𝜔)𝑥≥0,𝑥^𝑇𝑀(𝜔)𝑥=0=1⟹𝑥=0. (11) Moreover, Zhang and Chen [11] introduce a new concept of stochastic 𝑅₀ function, which can be regarded as a generalization of the stochastic 𝑅₀ matrix given in [8].

Throughout this paper, we suppose that the sample space Ω is nonempty and compact set and that the function 𝐹(𝑥,𝜔) is continuous with respect to 𝑥 and 𝜔. On the other hand, we will use the following notations: 𝐼(𝑥)={𝑖∶𝑥_𝑖=0} and 𝐽(𝑥)={𝑖∶𝑥_𝑖≠0} for a given vector 𝑥∈𝐑^𝑛. ⟨𝑙,𝑛⟩ represents the set {𝑙,𝑙+1,…,𝑙+𝑛} for natural numbers 𝑙 and 𝑢 with 𝑙<𝑢. 𝑥₊=max{𝑥,0} for any given vector 𝑥. ‖⋅‖ refers to the Euclidean norm.

The remainder of the paper is organized as follows: in Section 2, we introduce the concepts of a stochastic 𝑃-function, a stochastic 𝑃₀-function, and a stochastic uniformly 𝑃-function, which can be regarded as a generalization of the deterministic 𝑃, 𝑃₀-function, and uniformly 𝑃-function or an extension of stochastic 𝑃 matrix and stochastic 𝑃₀ matrix [14]. In addition, some properties of a stochastic 𝑃(𝑃₀)-function are given. In Section 3, we show the sufficient conditions for the solution set of ERM problem to be nonempty and bounded. In Section 4, we discuss error bounds of SCP(𝐹(𝑥,𝜔)). In Section 5, an algorithm will be given to solve ERM model. We then give conclusions in Section 6.

2. Stochastic 𝑃(𝑃₀)-Function

It is well known that the 𝑃-function, 𝑃₀-function, and uniformly 𝑃-function play an important role in the nonlinear complementarity problems theory [1]. We will introduce a new concept of stochastic 𝑃-function, 𝑃₀-function, and uniformly 𝑃-function, which can be regarded as a generalization of their deterministic form or stochastic 𝑃 matrix and stochastic 𝑃₀ matrix.

Definition 1 (see [14]). 𝑀(⋅) is called a stochastic 𝑃(𝑃₀)-matrix if there exists 𝑖∈𝐽(𝑥) such that, for every 𝑥≠0 in 𝐑^𝑛, 𝒫𝜔∶𝑥_𝑖(𝑀(𝜔)𝑥)_𝑖>0(≥0)>0. (12)

Definition 2. A function 𝐹∶𝐑^𝑛×Ω→𝐑^𝑛 is a stochastic 𝑃(𝑃₀)-function if there exist 𝑖∈𝐽(𝑥,𝑦), 𝑖∈⟨1,𝑛⟩ such that, for every 𝑥≠𝑦 in 𝐑^𝑛, 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)>0. (13)

Definition 3. A function 𝐹∶𝐑^𝑛×Ω→𝐑^𝑛 is a stochastic uniformly 𝑃-function if there exists a positive constant 𝛼 and 𝑖∈𝐽(𝑥,𝑦), 𝑖∈⟨1,𝑛⟩ such that, for every 𝑥≠𝑦 in 𝐑^𝑛, 𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≥𝛼𝑥−𝑦²>0. (14)

Clearly, every stochastic uniformly 𝑃-function must be a stochastic 𝑃-function, which in turn must be a stochastic 𝑃₀-function. We further cite the definition of “equicoercive" in [11]. More information about this definition can be found in [11].

Definition 4 (see [11]). We say that 𝐹∶𝐑^𝑛×Ω→𝐑^𝑛 is equicoercive on 𝒟⊆𝐑^𝑛, if, for any {𝑥^𝑘}⊆𝒟 satisfying ‖𝑥^𝑘‖→∞, the existence of {𝜔^𝑘}⊆suppΩ with lim_𝑘→∞𝐹_𝑖(𝑥^𝑘,𝜔^𝑘)=∞(lim_𝑘→∞(−𝐹_𝑖(𝑥^𝑘,𝜔^𝑘))₊=∞) for some 𝑖∈⟨1,𝑛⟩ implies that 𝒫𝜔∶lim_𝑘→∞𝐹_𝑖𝑥^𝑘𝒫,𝜔=∞>0𝜔∶lim_𝑘→∞−𝐹_𝑖𝑥^𝑘,𝜔₊=∞>0, (15) where suppΩ∶=𝜔∈Ω∶_{𝐵𝜔(𝜔,𝜈)∩Ω}𝑑𝐹(𝜔)>0forany𝜈>0 (16) and 𝐵_𝜔(𝜔,𝜈)∶={𝜔∶‖𝜔−𝜔‖<𝜈} and 𝐹(𝜔) is the distribution function of 𝜔.

More details about suppΩ were included in [8].

Proposition 5. If 𝐹 is a stochastic 𝑃₀-function, then 𝐹+𝜀𝑥 is a stochastic 𝑃-function for every 𝜀>0.

Proof. From the definition of stochastic 𝑃₀-function, there exist 𝑖∈𝐽(𝑥,𝑦), 𝑖∈⟨1,𝑛⟩ such that, for every 𝑥≠𝑦, 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≥0>0. (17) Hence, we have 𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)+𝜀𝑥_𝑖−𝐹_𝑖𝑦,𝜔+𝜀𝑦_𝑖𝑥>0=𝒫𝜔∶_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔+𝜀𝑥_𝑖−𝑦_𝑖²𝑥>0≥𝒫𝜔∶_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≥0>0. (18)

This proposition gives the relationship between stochastic 𝑃₀-function and stochastic 𝑃-function.

Proposition 6. Let 𝐹 be an affine function of 𝑥 for any 𝜔∈Ω defined by (2). Then 𝐹 is a stochastic 𝑃(𝑃₀)-function if and only if 𝑀(⋅) is a stochastic 𝑃(𝑃₀) matrix.

Proof. By the definition of stochastic 𝑃(𝑃₀)-function, we have that there exist 𝑖∈𝐽(𝑥,𝑦), 𝑖∈⟨1,𝑛⟩ such that, for every 𝑥≠𝑦, 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)>0, (19) which is equivalent to 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝑀(𝜔)𝑥−𝑦_𝑖>0(≥0)>0, (20) when 𝐹 is defined by (2). Set 𝑧=𝑥−𝑦; then 𝑧≠0, and we have that formulation (20) holds if and only if there exists 𝑖∈𝐽(𝑧) such that, for every 𝑧≠0, 𝒫𝜔∶𝑧_𝑖(𝑀(𝜔)𝑧)_𝑖>0(≥0)>0. (21) Hence, 𝑀(⋅) is a stochastic 𝑃(𝑃₀) matrix.

Proposition 7. 𝐹 is a stochastic 𝑃(𝑃₀)-function if and only if there exists a 𝜔∈suppΩ such that 𝐹(⋅,𝜔) is a 𝑃(𝑃₀)-function.

Proof. For the “if” part, suppose on the contrary that 𝐹 is not a stochastic 𝑃(𝑃₀)-function, and then there exist 𝑥,𝑦,𝑥≠𝑦 in 𝐑^𝑛 for any 𝑖∈⟨1,𝑛⟩ satisfying 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)=0. (22)
On the other hand, since 𝐹(⋅,𝜔) is a 𝑃(𝑃₀)-function, then for 𝑥,𝑦 there exist 𝑖∈𝐽(𝑥,𝑦), 𝑖∈⟨1,𝑛⟩ such that 𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0). (23) Notice that 𝜔∈suppΩ, by the definition of suppΩ in (16), we have 𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)>0. (24) This contradicts formulation (22). Therefore, 𝐹 is a stochastic 𝑃(𝑃₀)-function.
Now for the “only if” part, suppose on the contrary that there does not exist a 𝜔∈suppΩ such that 𝐹(⋅,𝜔) is a 𝑃(𝑃₀)-function. Then for any 𝑖∈⟨1,𝑛⟩, 𝜔∈suppΩ, there exists 𝑥,𝑦,𝑥≠𝑦 in 𝐑^𝑛, such that 𝑥_𝑖≠𝑦_𝑖,𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≤0(<0), (25) which means that 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∈suppΩ∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖(𝑦,𝜔>0≥0)=0. (26) By the definition of suppΩ in (16), we have 𝒫{𝜔∈Ω⧵suppΩ}=0. Hence, formulation (26) is equivalent to 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∈Ω∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)=0, (27) which contradicts definition (20). Therefore, there exists a 𝜔∈suppΩ such that 𝐹(⋅,𝜔) is a 𝑃(𝑃₀)-function.

Theorem 8. Suppose that 𝑓(𝑥)∶=𝐄[𝐹(𝑥,𝜔)] is a 𝑃(𝑃₀)-function. Then 𝐹 is a stochastic 𝑃(𝑃₀)-function.

Proof. Suppose on the contrary that 𝐹 is not a stochastic 𝑃(𝑃₀)-function, then there exist 𝑥,𝑦,𝑥≠𝑦 in 𝐑^𝑛 for any 𝑖∈⟨1,𝑛⟩ satisfying 𝑥_𝑖≠𝑦_𝑖,𝒫𝜔∶𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔>0(≥0)=0. (28) This means that 𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≤0(<0) (29) always holds for any 𝑖∈⟨1,𝑛⟩ and 𝜔∈Ω. Furthermore, following from (29), we have 𝐄𝑥_𝑖−𝑦_𝑖𝐹_𝑖(𝑥,𝜔)−𝐹_𝑖𝑦,𝜔≤0(<0), (30) that is 𝑥_𝑖−𝑦_𝑖𝑓_𝑖(𝑥)−𝑓_𝑖𝑦≤0(<0), (31) which contradicts the definition of 𝑃(𝑃₀)-function. Therefore, 𝐹 is a stochastic 𝑃(𝑃₀)-function.

Note that there is at most one solution (may not be a solution) for the EV model stochastic complementarity problems if 𝑓(𝑥)∶=𝐄[𝐹(𝑥,𝜔)] is a 𝑃(𝑃₀)-function.

3. Boundedness of Solution Set

Theorem 9. Suppose that 𝐹 is a stochastic uniformly 𝑃-function and 𝐹 is equicoercive on 𝐑^𝑛. Then the solution set of ERM model (7) defined by 𝜙_min and 𝜙_FB is nonempty and bounded.

Proof. Suppose on the contrary that the ERM model defined by 𝜙_min is not bounded. Thus there exist a sequence {𝑥^𝑘}⊂𝐑^𝑛₊ with ‖𝑥^𝑘‖→∞(𝑘→∞) and a constant 𝑐∈𝐑₊, such that 𝜃𝑥^𝑘≤𝑐,for∀𝑘. (32) Define the index set 𝐼⊆{1,…,𝑛} by 𝐼∶=𝑖∣𝑥^𝑘_𝑖isunbounded. (33) By assumption, we have 𝐼≠∅. We now define a sequence {𝑦^𝑘}⊆𝐑^𝑛 as follows: 𝑦^𝑘_𝑖𝑥∶=0if𝑖∈𝐼,_𝑘𝑖if𝑖∉𝐼. (34) From the definition of 𝑦^𝑘 and the fact that 𝐹 is a stochastic uniformly 𝑃-function, we obtain that for any 𝑥^𝑘,𝑦^𝑘, there exists 𝑖 such that 𝒫𝜔∶𝑥^𝑘_𝑖−𝑦^𝑘_𝑖𝐹_𝑖𝑥^𝑘,𝜔−𝐹_𝑖𝑦^𝑘𝑥,𝜔≥𝛼^𝑘−𝑦^𝑘2>0, (35) and hence there are 𝜔^𝑘∈suppΩ satisfying 𝑥^𝑘_𝑖−𝑦^𝑘_𝑖𝐹_𝑖𝑥^𝑘,𝜔^𝑘−𝐹_𝑖𝑦^𝑘,𝜔^𝑘≥𝛼𝑥^𝑘−𝑦^𝑘2. (36) Take subsequence 𝑥^𝑘𝑖,𝑦^𝑘𝑖 such that the corresponding subscript of (36) is 𝑗. Noting that 𝑗∈𝐼 and taking (36) into account, we have 𝛼_𝑗∈𝐼𝑥^𝑘𝑖_𝑗²≤𝑥^𝑘𝑖_𝑗𝐹_𝑗𝑥^𝑘𝑖,𝜔^𝑘𝑖−𝐹_𝑗𝑦^𝑘𝑖,𝜔^𝑘𝑖≤_𝑗∈𝐼𝑥^𝑘𝑖_𝑗²⋅_𝑗∈𝐼𝐹_𝑗𝑥^𝑘𝑖,𝜔^𝑘𝑖−𝐹_𝑗𝑦^𝑘𝑖,𝜔^𝑘𝑖, (37) from which we get 𝛼_𝑗∈𝐼𝑥^𝑘𝑖_𝑗²≤_𝑗∈𝐼𝐹_𝑗𝑥^𝑘𝑖,𝜔^𝑘𝑖−𝐹_𝑗𝑦^𝑘𝑖,𝜔^𝑘𝑖. (38) By definition, the sequence {𝑦^𝑘𝑖} remains bounded. From the continuity of 𝐹, it follows that the sequence {𝐹_𝑗(𝑦^𝑘𝑖,𝜔^𝑘𝑖)} is also bounded for every 𝑗∈𝐼. Hence, taking a limit in (38), we obtain that there is at least one index 𝑗∈𝐼 such that 𝑥^𝑘𝑖_𝑗⟶∞,𝐹_𝑗𝑥^𝑘𝑖,𝜔^𝑘𝑖⟶∞. (39) Since 𝐹 is equicoercive on 𝐑^𝑛, we have 𝒫𝜔∶lim_{𝑘𝑖→∞}𝐹_𝑗𝑥^𝑘𝑖,𝜔=∞>0. (40) Let Ω₁=𝜔∶lim_{𝑘𝑖→∞}min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔=∞. (41) Then 𝒫{Ω₁}>0. By Fatou’s Lemma [15], we have 𝐄_Ω1liminf_{𝑘𝑖→∞}min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔²≤liminf_{𝑘𝑖→∞}𝐄_Ω1min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔². (42) Since liminf_{𝑘𝑖→∞}min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔²=∞ (43) on Ω₁ and 𝒫{Ω₁}>0, then the left-hand side of formulation (42) is infinite. Therefore, liminf_{𝑘𝑖→∞}𝐄_Ω1min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔²=∞. (44) Moreover, it is easy to find 𝜃𝑥^𝑘𝑖=𝐄Φ(𝑥^𝑘𝑖,𝐹(𝑥^𝑘𝑖,𝜔))²≥𝐄_Ω1min𝑥^𝑘𝑖_𝑗,𝐹_𝑗𝑥^𝑘𝑖,𝜔²⟶∞ (45) as 𝑘_𝑖→∞. This contradicts formulation (32). Hence, the solution set of ERM model (7) defined by 𝜙_min is nonempty and bounded. Similar results about 𝜙_FB can be obtained by relation formulation (10).

4. Robust Solution

As we show, both EV model and ERM model give decisions by a deterministic formulation. However, the decisions may not be the best or may be even infeasible for each individual event. In fact, we should take risk into account to make a priori decision in many cases. Naturally, it is necessary to know how good or how bad the decision which we have given can be. In this section, we study the robustness of solutions of the ERM model. Let SOL(𝐹(𝑥,𝜔)) denote the solution set of SCP(𝐹(𝑥,𝜔)), and define the distance from a point 𝑥 to the set SOL(𝐹(𝑥,𝜔)) by dist(𝑥,SOL(𝐹(𝑥,𝜔)))∶=inf_{𝑥′∈SOL(𝐹(𝑥,𝜔))}𝑥−𝑥^′. (46)

Theorem 10. Assume that Ω={𝜔¹,𝜔²,…,𝜔^𝑁}⊂𝐑^𝑚, and 𝜔 takes values 𝜔¹,…,𝜔^𝑁 with respective probabilities 𝑝₁,…,𝑝_𝑁. Furthermore, suppose that for every 𝜔∈Ω, 𝐹(𝑥,𝜔) is uniformly P-function and Lipschitz continuous with respect to 𝑥. Then there is a constant 𝐶>0 such that 𝐄[dist(𝑥,SOL(𝐹(𝑥,𝜔)))]≤𝐶⋅𝜃(𝑥), (47) where 𝜃(𝑥) is defined by 𝜙_min or 𝜙_FB.

Proof. For any fixed 𝜔^𝑖, since 𝐹(𝑥,𝜔^𝑖) is uniformly 𝑃-function and Lipschitz continuous, from Corollary 3.19 of [16], we have unique solution 𝑥(𝜔^𝑖) of CP(𝐹(𝑥,𝜔^𝑖)), and there exists a constant 𝐶_𝑖 such that 𝑥−𝑥𝜔^𝑖≤𝐶_𝑖min𝑥,𝐹𝑥,𝜔^𝑖. (48) Letting 𝐶∶=((√2+2)/2)max{𝐶₁,…,𝐶_𝑁}, we have 𝐄²[dist(𝑥,SOL(𝐹(𝑥,𝜔)))]=𝐄²𝑥−𝑥𝜔^𝑖≤𝐄𝑥−𝑥𝜔^𝑖2≤^𝑁_𝑖=1𝑝_𝑖⋅𝐶²_𝑖min𝑥,𝐹𝑥,𝜔^𝑖2≤^𝑁_𝑖=1𝑝_𝑖⋅𝐶²_𝑖⋅√2+22²×^𝑛_𝑗=1𝐹²_𝑗𝑥,𝜔^𝑖+𝑥²_𝑗−𝐹_𝑗𝑥,𝜔^𝑖+𝑥_𝑗²≤𝐶^2𝑁_𝑖=1𝑝_𝑖^𝑛_𝑗=1𝐹²_𝑗𝑥,𝜔^𝑖+𝑥²_𝑗−𝐹_𝑗𝑥,𝜔^𝑖+𝑥_𝑗²=𝐶²𝜃(𝑥), (49) where the first inequality follows from Cauchy-Schwarz inequality, the second inequality follows from formulation (48), and the third inequality follows from formulation (10). This completes the proof of the theorem.

Theorem 10 particularly shows that for the solution 𝑥^∗ of (7), 𝐄dist𝑥^∗,SOL(𝐹(𝑥,𝜔))≤𝐶⋅𝜃(𝑥^∗). (50) This inequality indicates that the expected distance to the solution set SOL(𝐹(𝑥,𝜔)) for 𝜔∈Ω is also likely to be small at the solution 𝑥^∗ of (7). In other words, we may expect that a solution of the ERM formulation (7) has a minimum sensitivity with respect to random parameter variations in SCP(𝐹(𝑥,𝜔)). In this sense, solutions of (7) can be regarded as robust solutions for SCP(𝐹(𝑥,𝜔)).

5. Quasi-Monte Carlo and Derivative-Free Methods for Solving ERM Model

Note that the ERM model (7) included an expectation function, which is generally difficult to be evaluated exactly. Hence in this section, we first employ a quasi-Monte Carlo method to obtain approximation problems of (7) for numerical integration. Then, we consider derivative-free methods to solve these approximation problems.

By the quasi-Monte Carlo method, we obtain the following approximation problem of (7): min_{𝑥∈𝐑𝑛+}𝜃^𝑁(𝑥)1∶=𝑁_{𝜔𝑖∈Ω𝑁}Φ𝑥,𝜔^𝑖2𝜌𝜔^𝑖, (51) where Ω_𝑁={𝜔^𝑖∣𝑖=1,2,…,𝑁} is a set of observations generated by a quasi-Monte Carlo method such that Ω_𝑁⊆Ω and 𝜌(𝜔) stands for the probability density function. In the rest of this paper, we assume that the probability density function 𝜌 is continuous on Ω. For each 𝑁, 𝜃^𝑁(𝑥) is continuously differentiable function. We denote by 𝑥^𝑁 the optimal solutions of approximation problems (51). We are interested in the situation where the first-order derivatives of 𝜃^𝑁(𝑥) cannot be explicitly calculated or approximated.

Condition 1. Given a point 𝑥₀≥0, the level set 𝐿∶=𝑥≥0∣𝑓(𝑥)≤𝑓𝑥₀ (52) is compact.

Condition 2. If {𝑥^𝑁_𝑘} and {𝑦^𝑁_𝑘} are sequences of points such that 𝑥^𝑁_𝑘≥0, 𝑦^𝑁_𝑘≥0 converging to some 𝑥^𝑁 and 𝐼^𝑁_𝑘⊆𝐼(𝑥^𝑁)∶={𝑖∣𝑥^𝑁_𝑖=0} for all 𝑘, then dist𝑇_𝐼^𝑁_𝑘𝑥^𝑁_𝑘,𝑇_𝐼^𝑁_𝑘𝑦^𝑁_𝑘⟶0, (53) where dist(𝑇₁,𝑇₂)=max_{𝑑1∈𝑇1∶‖𝑑1‖=1}{min_{𝑑2∈𝑇2}‖𝑑₁−𝑑₂‖} and 𝑇_𝐼^𝑁_𝑘(𝑥)∶={𝑑^𝑁_𝑘∈𝐑^𝑛∣𝑑^𝑁_𝑘,𝑖≥0,∀𝑖∈𝐼^𝑁_𝑘}.

Condition 3. For every 𝑥^𝑁≥0 there exist scalars 𝛿>0, and 𝜂>0 such that min_𝑧≥0‖𝑧−𝑥‖≤𝜂^𝑛_𝑖=1max−𝑥_𝑖,0,∀𝑥∈𝑥∈𝐑^𝑛∣‖𝑥−𝑥‖≤𝛿. (54)

Condition 4. Given 𝑥^𝑁_𝑘 and 𝜖^𝑁_𝑘>0, the set of search directions 𝐷^𝑁_𝑘=𝑑_𝑘^𝑁,𝑗,𝑗=1,…,𝑟^𝑁_𝑘,with𝑑_𝑘^𝑁,𝑗=1, (55) satisfing 𝑟^𝑁_𝑘 is uniformly bounded and cone{𝐷^𝑁_𝑘}=𝑇(𝑥^𝑁_𝑘;𝜖^𝑁_𝑘). Here, cone𝐷^𝑁_𝑘=𝑑_𝑘^𝑁,1𝛽¹+⋯+𝑑^{𝑁,𝑟𝑁𝑘}_𝑘𝛽^𝑟𝑁𝑘∶𝛽¹≥0,…,𝛽^𝑟𝑁𝑘,𝑇𝑥≥0_𝑁𝑘;𝜖^𝑁_𝑘=𝑑^𝑁_𝑘∈𝐑^𝑛∣𝑑^𝑁_𝑘,𝑖≥0,𝑥^𝑁_𝑘,𝑖≤𝜖^𝑁_𝑘. (56)

Under Conditions 1, 2, and 3 and by choosing 𝐷^𝑁_𝑘 satisfying Condition 4 with 𝜖^𝑁_𝑘→0, then the following generated iterates have at least one cluster point that is a stationary point of (51) for each 𝑁.

Algorithm 11. Parameters: 𝑥^𝑁₀≥0, 𝛼^𝑁₀>0, 𝛾^𝑁>0, 𝜃^𝑁₁∈(0,1), 𝜃^𝑁₂∈(0,1), 𝜖^𝑁₀>0.
Step  1. Set 𝑘^𝑁=0.
Step  2. Choose a set of directions 𝐷^𝑁_𝑘={𝑑_𝑘^𝑁,𝑗,𝑗=1,…,𝑟^𝑁_𝑘} satisfying Condition 4.
Step  3. (a) Set 𝑗=1, 𝑦_𝑘^𝑁,𝑗=𝑥^𝑁_𝑘, 𝛼_𝑘^𝑁,𝑗=𝛼^𝑁_𝑘. (b) Compute the maximum stepsize 𝛼_𝑘^𝑁,𝑗 such that 𝑦^𝑁,𝑗_𝑖,𝑘+𝛼_𝑘^𝑁,𝑗𝑑^𝑁,𝑗_𝑖,𝑘≥0 for all 𝑖. Set 𝛼_𝑘^𝑁,𝑗=min{𝛼_𝑘^𝑁,𝑗,𝛼_𝑘^𝑁,𝑗}. (c) If 𝛼_𝑘^𝑁,𝑗>0 and 𝜃^𝑁(𝑦_𝑘^𝑁,𝑗)≤𝜃^𝑁(𝑦_𝑘^𝑁,𝑗)−𝛾(𝛼_𝑘^𝑁,𝑗)², set 𝛼_𝑘^𝑁,𝑗+1=𝛼_𝑘^𝑁,𝑗; otherwise set 𝛼_𝑘^𝑁,𝑗=0,𝑦_𝑘^𝑁,𝑗+1=𝑦_𝑘^𝑁,𝑗,𝛼_𝑘^𝑁,𝑗+1=𝜃^𝑁₁𝛼_𝑘^𝑁,𝑗. (d) If 𝛼_𝑘^𝑁,𝑗=𝛼_𝑘^𝑁,𝑗, set 𝜖^𝑁_𝑘+1=𝜖^𝑁_𝑘, and go to Step 4. (e) If 𝑗<𝑟^𝑁_𝑘, set 𝑗=𝑗+1, and go to Step 3(b). Otherwise set 𝜖^𝑁_𝑘+1=𝜃^𝑁₂𝜖^𝑁_𝑘 and go to Step 4.
Step  4. Find 𝑥^𝑁_𝑘+1≥0 such that 𝜃^𝑁(𝑥^𝑁_𝑘+1)≤𝜃^𝑁(𝑦_𝑘^𝑁,𝑗+1). Set 𝛼^𝑁_𝑘+1=𝛼_𝑘^𝑁,𝑗+1,𝑟_𝑘̃=𝑗,𝑘=𝑘+1, and go to Step 2.

For this algorithm, it is easy to proof that if 𝑥^𝑁_𝑘 is the sequence produced by algorithm under Conditions 1–4, then 𝑥^𝑁_𝑘 is bounded and there exists at least one cluster point which is a stationary point of problem (51) for each 𝑁.

6. Conclusions

The SCP(𝐹(𝑥,𝜔)) has a wide range of applications in engineering and economics. Therefore, it is meaningful and interesting to study this problem. In this paper, we give the definitions of stochastic 𝑃-function, stochastic 𝑃₀-function and stochastic uniformly 𝑃-function, which can be regarded as a generalization of the deterministic formulation or an extension of a stochastic 𝑅₀ function given in [11]. Moreover, we consider the conditions when the function is a stochastic 𝑃(𝑃₀)-function. Furthermore, we show that the involved function being a stochastic uniformly 𝑃-function and equi-coercive [11] are sufficient conditions for the solution set of the expected residual minimization problem to be nonempty and bounded. Finally, we illustrate that the ERM formulation produces robust solutions with minimum sensitivity in violation of feasibility with respect to random parameter variations in SCP(𝐹(𝑥,𝜔)). On the other hand, we employ a quasi-Monte Carlo method to obtain approximation problems of (7) for dealing numerical integration and further consider derivative-free methods to solve these approximation problems.

Acknowledgments

This work was supported by NSFC Grants no. 11226238 and no. 11226230, and predeclaration fund of state project of Liaoning university 2012, 2012LDGY01, and University Scientific Research Projects of School of Education Department of Liaoning Province 2012, 2012427.