Quantitative Stability of Optimization Problems with Stochastic Constraints

Abstract

In this paper, we consider optimization problems with stochastic constraints. We derive quantitative stability results for the optimal value function, the optimal solution set and the feasible solution set of optimization models in which the underlying stochastic constraints involve the mathematical expectation of random single-valued and set-valued mappings, respectively. New primal sufficient conditions are developed for the uniform error bound property of the stochastic constraint system for the single-valued case.

1. Introduction

Consider the following optimization problem with stochastic constraint (OPSC):

$$\begin{array}{cc}\hfill & \underset{x\in X}{min}f\left(x\right)\hfill \\ \hfill & s.t.{\mathbb{E}}_P\left[\gamma (x,\xi )\right]\le 0,\hfill \end{array}$$

(1)

where $f:{\mathbb{R}}^n\to \mathbb{R}$ is a deterministic function, $X\subset {\mathbb{R}}^n$ is the direct constraint set for the decision vector $x\in {\mathbb{R}}^n$, $\gamma :{\mathbb{R}}^n\times \mathsf{\Xi }\to \mathbb{R}$, $\xi :\mathsf{\Omega }\to \mathsf{\Xi }$ is a random variable defined on the probability space $(\mathsf{\Omega },\mathcal{F},P)$ with support set $\mathsf{\Xi }\subset {\mathbb{R}}^m$ and probability distribution P, and ${\mathbb{E}}_P[·]$ denotes the expected value with respect to P.

Model (1) can be viewed as an extension of the classic deterministic mathematical programming problem. It is closely related to the distributionally robust optimization model, as well as optimization problems with stochastic dominance constraints, which have received considerable attention over the past decade for their theoretical importance and extensive applications (for more details, please refer to references [1,2,3,4,5,6,7,8,9,10] and the references therein).

The probability distributiton of random variables in traditional stochastic programs is often assumed complete knowledge. However, in many practical applications, it is usually impossible or expensive for us to know the true probability distribution, while we are able to obtain some partial information, such as regarding prior moments, samples of the random variables, or the historical data, and use them to construct an ambiguity set of probability distributions which contain or approximate the true probability distribution (see references [11,12,13,14,15,16,17,18] and the references therein). This motivates us to consider the following perturbed optimization model:

$$\begin{array}{cc}\hfill & \underset{x\in X}{min}f\left(x\right)\hfill \\ \hfill & s.t.{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\le 0,\hfill \end{array}$$

(2)

where Q is taken from the ambiguity set for the possible probability distributions of the random vector $\mathsf{\Xi }\subset {\mathbb{R}}^m$ on some probability space $(\mathsf{\Omega },\mathcal{F},P)$.

It is of great importance to investigate the quantitative stability of OPSC by analyzing the impact of the variation of probability distributions on the optimal value, the optimal solution set and the feasible solution set. To this end, upon utilizing the Ekeland’s Variational Principle, one of the major tasks of this paper is to develop new primal sufficient conditions for the uniform error bound property of the inequality constraint system in (2), which enables us to derive stability estimation for OPSC (2) as Q approximates P under appropriate metrics. The uniform error bound property plays an essential role in the proof of the main results. For more information regarding the error bounds of deterministic systems, please refer to references [19,20] and the references therein.

OPSC (1) can be viewed as a particular case of the following optimization problem constrained by a stochastic generalized equation (OPSGE):

$$\begin{array}{cc}\hfill & \underset{x\in X}{min}f\left(x\right)\hfill \\ \hfill & s.t.0\in {\mathbb{E}}_P[\Gamma (x,\xi )]+\mathcal{G}\left(x\right),\hfill \end{array}$$

(3)

where $X\subset {\mathbb{R}}^n$, $f:{\mathbb{R}}^n\to \mathbb{R},\Gamma :\mathcal{X}\times \mathsf{\Xi }\rightrightarrows \mathcal{Y}$ and $\mathcal{G}:\mathcal{X}\rightrightarrows \mathcal{Y}$ are closed set-valued mappings, $\mathcal{X}$ and $\mathcal{Y}$ are subsets of ${\mathbb{R}}^n$ and ${\mathbb{R}}^d$, respectively, $\xi :\mathsf{\Omega }\to \mathsf{\Xi }$ is a random vector defined on a probability space $(\mathsf{\Omega },\mathcal{F},P)$ with support set $\mathsf{\Xi }\subset {\mathbb{R}}^m$ and probability distribution P, and ${\mathbb{E}}_P[·]$ denotes the expected value with respect to P, that is,

$$\begin{array}{cc}\hfill {\mathbb{E}}_P[\Gamma (x,\xi )]& :={\int }_{\mathsf{\Xi }}\Gamma (x,\xi )dP\left(\xi \right)\hfill \\ & =\left\{{\int }_{\mathsf{\Xi }}\psi \left(\xi \right)P\left(d\xi \right):\psi \mathrm{is}\mathrm{a}\mathrm{Bochner}\mathrm{integrable}\mathrm{selection}\mathrm{of}\Gamma (x,·)\right\}.\hfill \end{array}$$

The expected value of $\Gamma$ is widely known as Aumann’s integral of the set-valued mapping (for more details, please refer to references [21,22,23]). Indeed, if we set $d=1$ and $\mathcal{G}\left(x\right)=[0,+\infty )$ (for all $x\in \mathcal{X}$), then $\Gamma$ is single-valued and OPSGE (3) reduces to OPSC (1).

The stochastic generalized equation (SGE) formulation in model (3) is a natural extension of the deterministic generalized equation [24]. It provides a uniform platform for describing the first order optimality/equilibrium conditions of nonsmooth stochastic optimization problems, stochastic equilibrium problems and stochastic games (see references [15,16] and the references therein). In particular, when $\Gamma$ is single-valued and $\mathcal{G}\left(x\right)$ is a normal cone of a set, SGE in (3) is known as a stochastic variational inequality which has been studied extensively over the past few years (see references [11,18] and the references therein).

Another major objective of this paper is to establish the quantitative stability result for OPSGE (3) by estimating the variation of the feasible solution set and the optimal value function, respectively, as the underlying probability distribution P perturbs from an ambiguity set. The obtained result is a supplement to and extension of the existing ones in the literature and the proof is completely self-contained.

The rest of the paper is structured as follows. Section 2 contains definitions of the basic properties under consideration and some preliminary results used throughout the following sections. In Section 3, we conduct stability analysis for both OPSC (1) and OPSGE (3) under the perturbation of the underlying probability distribution. Estimations for the variation of the optimal value function, the optimal solution set and the feasible solution set of the aforementioned optimization models are obtained.

2. Preliminaries

Throughout this paper, we use the following notations. Let $\mathbb{N}$ and $\mathbb{R}$ denote the natural number set and real number set, respectively. Let $\parallel ·\parallel$ denote the Euclidean norm in ${\mathbb{R}}^n$ and ${\left[a\right]}_{+}:=max\{a,0\}$, for all $a\in \mathbb{R}$. The symbol $B(x,r)$ indicates the closed ball of radius $r>0$ centered at $x\in {\mathbb{R}}^n$. Given subsets $C,D\subset {\mathbb{R}}^n$, define the distance from $x\in X$ to C by

$$d(x,C):=inf\{\parallel x-c\parallel \mid c\in C\}.$$

The excess from C to D is defined by

$$\mathbb{D}(C,D):=sup\left\{d\right(c,D)\mid c\in C\}.$$

The Pompeiu–Hausdorff distance between C and D is defined as follows:

$$\mathbb{H}(C,D):=max\left\{\mathbb{D}\right(C,D),\mathbb{D}(D,C\left)\right\}.$$

We use the convention that $d(x,\varnothing ):=\infty$, $\mathbb{D}(\varnothing ,D):=0$, if $D\ne \varnothing$ and $\mathbb{D}(\varnothing ,D):=\infty$, if $D=\varnothing$. Let $\mathcal{P}\left(\mathsf{\Omega }\right)$ denote the set of all Borel probability measures on $\mathsf{\Omega }$ and $\mathcal{P}\subset \mathcal{P}\left(\mathsf{\Omega }\right)$ represent the ambiguity set for possible probability distributions of P.

Let $F:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ be a set-valued mapping and its domain and graph be denoted, respectively, by

$$\mathrm{dom}\left(F\right):=\{x\in {\mathbb{R}}^n\mid F\left(x\right)\ne \varnothing \}\mathrm{and}\mathrm{gph}\left(F\right):=\{(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}^m\mid y\in F\left(x\right)\}.$$

The symbol $F^{-1}:{\mathbb{R}}^m\rightrightarrows {\mathbb{R}}^n$ stands for the inverse mapping of F with $F^{-1}\left(y\right)=\{x\in {\mathbb{R}}^n:y\in F\left(x\right)\}$ for all $y\in {\mathbb{R}}^m$.

Next, we recall the notions of metric regularity and Lipschitz continuity of set-valued mappings (cf. reference [25]) which will be employed in our study.

Definition 1.

Let $X\subset {\mathbb{R}}^n,Y\subset {\mathbb{R}}^m$ and $\mathsf{\Phi }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ be a set-valued mapping. If there exists $\kappa >0$ such that

$$d(x,{\mathsf{\Phi }}^{-1}\left(y\right))\le \kappa d(y,\mathsf{\Phi }\left(x\right)),\forall x\in Xandy\in Y,$$

(4)

then we say that Φ is metrically regular on $X\times Y$ with constant κ.

Definition 2.

Let $X\subset {\mathbb{R}}^n,Y\subset {\mathbb{R}}^m$ and $\mathsf{\Phi }:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ be a set-valued mapping. We say that Φ is Lipschitz continuous on X with constant $\iota >0$, if

$$\mathbb{H}(\mathsf{\Phi }\left(x\right),\mathsf{\Phi }\left(x^{\prime }\right))\le \iota \parallel x-x^{\prime }\parallel ,\forall x,x^{\prime }\in X.$$

(5)

Let us finish this section by recalling the well-known Ekeland Variational Principle (cf. reference [26]) which plays a fundamental role in the proofs of our main results.

Lemma 1

(Ekeland Variational Principle). Let X be a complete metric space and $f:X\to \mathbb{R}\cup \{+\infty \}$ be a proper lower semicontinuous function. Let $\epsilon >0$ and $\overline{x}\in X$ such that

$$f\left(\overline{x}\right)\le {\inf }_{x\in X}f\left(x\right)+\epsilon .$$

Then, for any $\lambda >0$ there exists $x\in X$ such that

• $d(x,\overline{x})\le \lambda ,$
• $f\left(x\right)\le f\left(\overline{x}\right),$
• $f\left(x\right)<f\left(u\right)+\frac{\epsilon }{\lambda }d(u,x)$ for all $u\in X\setminus \left\{x\right\}$.

3. Stability Analysis of Optimization Problems with Stochastic Constraints

In this section, we aim to investigate the variation of the optimal value function, the optimal solution set and the feasible solution set of optimization problem with stochastic constraints, respectively, when the probability measure is perturbed. The obtained results are twofold. We provide stability analysis for both optimization model (2) and model (3), which cover the cases of single-valued and set-valued SGE constraints.

3.1. Stability Analysis of OPSC (2)

Recall that in optimization model (2), we are looking for candidates $x\in X$ such that

$${\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\le 0,$$

(6)

where $X\subset {\mathbb{R}}^n$ is the direct constraint set for the decision vector $x\in {\mathbb{R}}^n$ and Q is a perturbation of the probability distribution P of the random vector $\xi \in {\mathbb{R}}^m$.

To characterize the variation of probability measures with respect to OPSC (2), we define the following pseudometric induced by $\mathcal{H}$:

$$\mathcal{D}(P,Q):=\underset{h\in \mathcal{H}}{sup}|{\mathbb{E}}_P\left[h\left(\xi \right)\right]-{\mathbb{E}}_Q\left[h\left(\xi \right)\right]|,\forall P,Q\in \mathcal{P},$$

(7)

where $\mathcal{H}$ is a set of random functions:

$$\mathcal{H}:=\left\{h\right(·)\mid h(\xi )=\gamma (x,\xi )\mathrm{for}x\in X\}.$$

For convenience, let us denote by $\mathbb{S}\left(Q\right)$ the feasible solution set of (2), i.e.,

$$\mathbb{S}\left(Q\right):=\{x\in X\mid {\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\le 0\}.$$

Furthermore, let $S\left(Q\right)$ and $\vartheta \left(Q\right)$ denote the optimal solution set and the optimal value of model (2), respectively.

Definition 3.

We say that system (6) has a uniform error bound with respect to the constraint set X and the ambiguity set $\mathcal{P}\subset \mathcal{P}\left(\mathsf{\Omega }\right)$, if there exists $\kappa >0$ such that

$$d(x,\mathbb{S}\left(Q\right))\le \kappa {\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+},\forall (x,Q)\in X\times \mathcal{P}.$$

To start with, we establish the following sufficient condition for the uniform error bound property of constraint system (6) by utilizing the Ekeland’s Variational Principle, which will be beneficial for the proof of our main results.

Proposition 1.

Suppose that the mapping $x\mapsto \gamma (x,\xi )$ is lower semicontinuous for any $\xi \in {\mathbb{R}}^m$. Let $\tau \in (0,+\infty )$ be such that, for any $x\in X$ and $Q\in \mathcal{P}$ with ${\mathbb{E}}_Q\left[\gamma (x,\xi )\right]>0$, there exists $x^{\prime }\in X\setminus \left\{x\right\}$, satisfying

$${\left[\gamma (x^{\prime },·)\right]}_{+}\le \gamma (x,·)-\tau \parallel x^{\prime }-x\parallel .$$

(8)

Then, we have $\mathrm{dom}\left(\mathbb{S}\right)=\mathcal{P}$ and

$$d(x,\mathbb{S}\left(Q\right))\le \frac{1}{\tau }{\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+},\forall (x,Q)\in X\times \mathcal{P}.$$

(9)

Consequently, system (6) has a uniform error bound with respect to X and $\mathcal{P}$.

Proof. 

Note that $\gamma (·,\xi (\omega \left)\right)$ is lower semicontinuous; it follows from reference [27], (Theorem 7.42) that for any $Q\in \mathcal{P}$, the mapping $x\mapsto {\mathbb{E}}_Q\left[\gamma (x,\xi )\right]$ is also lower semicontinuous. Using (8), we know that for any $x\in X$ and $Q\in \mathcal{P}$ with ${\mathbb{E}}_Q\left[\gamma (x,\xi )\right]>0$, there exists $x^{\prime }\in X\setminus \left\{x\right\}$ such that

$${\mathbb{E}}_Q\left[{\left[\gamma (x^{\prime },\xi )\right]}_{+}\right]\le {\mathbb{E}}_Q\left[\gamma (x,\xi )\right]-\tau \parallel x^{\prime }-x\parallel .$$

(10)

For any $Q\in \mathcal{P}$, we claim that $\mathbb{S}\left(Q\right)\ne \varnothing$. Indeed, if this is not the case, we have

$${\mathbb{E}}_Q\left[\gamma (x,\xi )\right]>0,\forall x\in X.$$

(11)

To proceed, we pick a sequence $\left\{x_n\right\}\subset X$ such that

$${\mathbb{E}}_Q\left[\gamma (x_n,\xi )\right]<\underset{u\in X}{inf}{\mathbb{E}}_Q\left[\gamma (u,\xi )\right]+\frac{1}{n^2},\forall n\in \mathbb{N}.$$

Applying Lemma 1 to the function $x\mapsto {\mathbb{E}}_Q\left[\gamma (x,\xi )\right]$, we are able to find a sequence $\left\{x_n^{\prime }\right\}\subset X$ such that $\parallel x_n-x_n^{\prime }\parallel <\frac{1}{n}$ and

$${\mathbb{E}}_Q\left[\gamma (x_n^{\prime },\xi )\right]<{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]+\frac{1}{n}\parallel x-x_n^{\prime }\parallel ,\forall x\in X\setminus \left\{x_n^{\prime }\right\}.$$

This is a contradiction to (10) and (11). Therefore, $\mathbb{S}\left(Q\right)\ne \varnothing$, and hence $\mathrm{dom}\left(\mathbb{S}\right)=\mathcal{P}$.

Now we are ready to show that (9) holds. To this end, assume, to the contrary, that there exists $(x_0,Q_0)\in X\times \mathcal{P}$ such that

$${\left[{\mathbb{E}}_{Q_0}\left[\gamma (x_0,\xi )\right]\right]}_{+}<\tau d(x_0,\mathbb{S}\left(Q_0\right)).$$

Then, $d(x_0,\mathbb{S}\left(Q_0\right))>0$, which indicates that ${\mathbb{E}}_{Q_0}\left[\gamma (x_0,\xi )\right]>0$. Let ${\tau }^{\prime }\in (0,\tau )$ be close enough to $\tau$ such that

$${\mathbb{E}}_{Q_0}\left[\gamma (x_0,\xi )\right]<\underset{u\in X}{inf}{\left[{\mathbb{E}}_{Q_0}\left[\gamma (u,\xi )\right]\right]}_{+}+{\tau }^{\prime }d(x_0,\mathbb{S}\left(Q_0\right)).$$

(12)

Applying Lemma 1 to the function $x\mapsto {\left[{\mathbb{E}}_{Q_0}\left[\gamma (x,\xi )\right]\right]}_{+}$ again, we obtain $x_0^{\prime }\in X$ such that $\parallel x_0-x_0^{\prime }\parallel <\frac{{\tau }^{\prime }}{\tau }d(x_0,\mathbb{S}\left(Q_0\right))$ and

$${\left[{\mathbb{E}}_{Q_0}\left[\gamma (x_0^{\prime },\xi )\right]\right]}_{+}<{\left[{\mathbb{E}}_{Q_0}\left[\gamma (x,\xi )\right]\right]}_{+}+\tau \parallel x-x_0^{\prime }\parallel ,\forall x\in X\setminus \left\{x_0^{\prime }\right\}.$$

Together with (10), we conclude that ${\mathbb{E}}_{Q_0}\left[\gamma (x_0^{\prime },\xi )\right]=0$, and then $x_0^{\prime }\in \mathbb{S}\left(Q_0\right)$, which is a contradiction to the fact that

$$\parallel x_0-x_0^{\prime }\parallel <\frac{{\tau }^{\prime }}{\tau }d(x_0,\mathbb{S}\left(Q_0\right))<d(x_0,\mathbb{S}\left(Q_0\right)).$$

Hence, (9) holds true. □

Without the semicontinuity assumption, we next provide a different primal condition to ensure the uniform error bound property of the system (6).

Proposition 2.

Let $\rho \in [0,1)$ and $\tau \in (0,+\infty )$ be constants. Suppose that for each $Q\in \mathcal{P}$ and $x\in X\setminus \mathbb{S}\left(Q\right)$, there exists $x^{\prime }\in X\setminus \left\{x\right\}$ such that

$$d(x^{\prime },\mathbb{S}\left(Q\right))\le \rho d(x,\mathbb{S}\left(Q\right))$$

(13)

and

$${\left[\gamma (x^{\prime },·)\right]}_{+}\le \gamma (x,·)-\tau \parallel x^{\prime }-x\parallel .$$

(14)

Then, we conclude that (9) holds.

Proof. 

To show the validity of (9), we fix any $Q\in \mathcal{P}$ and $x\in X$. If $x\in \mathbb{S}\left(Q\right)$, then (9) holds automatically, hence we may assume that $x\notin \mathbb{S}\left(Q\right)$. Let $x_0=x$, and suppose that $x_0,\dots ,x_k$ are selected from X such that

$$d(x_i,\mathbb{S}\left(Q\right))\le {\rho }^{i}d(x,\mathbb{S}\left(Q\right))\mathrm{and}\tau \parallel x_{i-1}-x_i\parallel \le \gamma (x_{i-1},·)-{\left[\gamma (x_i,·)\right]}_{+},\forall 1\le i\le k.$$

If $x_k\in \mathbb{S}\left(Q\right)$, we set $x_{k+1}=x_k$. Otherwise, by inequalities (13) and (14), there exists $x_{k+1}\in X\setminus \left\{x_k\right\}$ such that

$$d(x_{k+1},\mathbb{S}\left(Q\right))\le \rho d(x_k,\mathbb{S}\left(Q\right))\le {\rho }^{k+1}d(x,\mathbb{S}\left(Q\right))$$

and

$$\tau \parallel x_k-x_{k+1}\parallel \le \gamma (x_k,·)-{\left[\gamma (x_{k+1},·)\right]}_{+}.$$

Hence, inductively we obtain a sequence $\left\{x_n\right\}$ such that $x_0=x$ and

$$d(x_n,\mathbb{S}\left(Q\right))\le {\rho }^{n}d(x,\mathbb{S}\left(Q\right))\mathrm{and}\tau \parallel x_{n-1}-x_n\parallel \le \gamma (x_{n-1},·)-{\left[\gamma (x_n,·)\right]}_{+},\forall n\in \mathbb{N}.$$

Therefore,

$${\mathbb{E}}_Q\left[{\left[\gamma (x_n,\xi )\right]}_{+}\right]\le {\mathbb{E}}_Q\left[\gamma (x_{n-1},\xi )\right]-\tau \parallel x_{n-1}-x_n\parallel ,\forall n\in \mathbb{N}.$$

(15)

To proceed, we divide our proof into two cases. (i) Suppose that there exist elements of the sequence $\left\{x_n\right\}$ which are contained in $\mathbb{S}\left(Q\right)$. For convenience, we set the first term as $x_{n_0}$. Then $\gamma (x_n,\xi )>0$ for all $n=1,2,\cdots ,n_0-1$ and ${\left[\gamma (x_{n_0},\xi )\right]}_{+}=0$. Hence,

$$\begin{array}{cc}\hfill \tau d(x,\mathbb{S}(Q\left)\right)& \le \tau \parallel x-x_{n_0}\parallel \le \tau \sum _{i=0}^{n_0-1}\parallel x_i-x_{i+1}\parallel \hfill \\ & \le \sum _{i=0}^{n_0-1}({\mathbb{E}}_Q\left[\gamma (x_i,\xi )\right]-{\mathbb{E}}_Q\left[\gamma (x_{i+1},\xi )\right])\hfill \\ & \le {\mathbb{E}}_Q\left[\gamma (x_0,\xi )\right]={\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+}.\hfill \end{array}$$

This shows that (9) holds. (ii) Assume that $x_n\notin \mathbb{S}\left(Q\right)$ for all $n\in \mathbb{N}$. Then, we have $\gamma (x_n,\xi )>0$, for any $n\in \mathbb{N}$. Note that $x_0=x$; it follows from inequality (15) that

$$\begin{array}{cc}\hfill \tau d(x,\mathbb{S}(Q\left)\right)& \le \tau d(x_n,\mathbb{S}\left(Q\right))+\tau \parallel x-x_n\parallel \hfill \\ \hfill & \le \tau {\rho }^{n}d(x,\mathbb{S}\left(Q\right))+\tau \sum _{i=0}^{n-1}\parallel x_i-x_{i+1}\parallel \hfill \\ & \le \tau {\rho }^{n}d(x,\mathbb{S}\left(Q\right))+\sum _{i=0}^{n-1}({\mathbb{E}}_Q\left[\gamma (x_i,\xi )\right]-{\mathbb{E}}_Q\left[\gamma (x_{i+1},\xi )\right])\hfill \\ & \le \tau {\rho }^{n}d(x,\mathbb{S}\left(Q\right))+{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\hfill \\ & =\tau {\rho }^{n}d(x,\mathbb{S}\left(Q\right))+{\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+}.\hfill \end{array}$$

Then,

$$d(x,\mathbb{S}\left(Q\right))\le \frac{1}{\tau (1-{\rho }^n)}{\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+},\forall n\in \mathbb{N}.$$

Letting $n\to \infty$, we conclude that (9) holds, which completes the proof. □

With the help of the uniform error bound property established in Proposition 1, we obtain the Lipschitz continuity of the feasible solution set as follows when the probability measure varies.

Theorem 1.

Assume that the conditions in Proposition 1 hold. Then,

$$\mathbb{H}(\mathbb{S}\left(Q^{\prime }\right),\mathbb{S}\left(Q\right))\le \frac{1}{\tau }\mathcal{D}(Q^{\prime },Q),\forall Q^{\prime },Q\in \mathcal{P}.$$

(16)

Proof. 

According to the assumptions, it follows from Proposition 1 that, for any $Q\in \mathcal{P}$, we have $\mathbb{S}\left(Q\right)\ne \varnothing$ and (9) holds. Pick any $Q,Q^{\prime }\in \mathcal{P}$ and $x\in \mathbb{S}\left(Q^{\prime }\right)$, then ${\mathbb{E}}_{Q^{\prime }}\left[\gamma (x,\xi )\right]\le 0$ and

$$\begin{array}{cc}\hfill d(x,\mathbb{S}(Q\left)\right)& \le \frac{1}{\tau }{\left[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]\right]}_{+}\le \frac{1}{\tau }{[{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]-{\mathbb{E}}_{Q^{\prime }}\left[\gamma (x,\xi )\right]]}_{+}\hfill \\ & \le \frac{1}{\tau }|{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]-{\mathbb{E}}_{Q^{\prime }}\left[\gamma (x,\xi )\right]|\hfill \\ & \le \frac{1}{\tau }\underset{x\in X}{sup}|{\mathbb{E}}_Q\left[\gamma (x,\xi )\right]-{\mathbb{E}}_{Q^{\prime }}\left[\gamma (x,\xi )\right]|\hfill \\ & =\frac{1}{\tau }\mathcal{D}(Q,Q^{\prime }),\hfill \end{array}$$

which indicates that $\mathbb{D}(\mathbb{S}\left(Q\right),\mathbb{S}\left(Q^{\prime }\right))\le \frac{1}{\tau }\mathcal{D}(Q,Q^{\prime })$. Note that Q and $Q^{\prime }$ are arbitrarily chosen from $\mathcal{P}$, we also have $\mathbb{D}(\mathbb{S}\left(Q^{\prime }\right),\mathbb{S}\left(Q\right))\le \frac{1}{\tau }\mathcal{D}(Q,Q^{\prime })$. Hence, (16) holds true. □

Now we are ready to state the first main result of this section. To this end, we define the growth function of problem (1) by ${\psi }_P:{\mathbb{R}}_{+}\to \mathbb{R}$, i.e.,

$${\psi }_P\left(t\right):=min\{f\left(x\right)-\vartheta \left(P\right)\mid d(x,S\left(P\right))\ge t,x\in \mathbb{S}\left(P\right)\},\forall t\in {\mathbb{R}}_{+}.$$

Its inverse function is ${\psi }_P^{-1}\left(s\right):=sup\{t\in {\mathbb{R}}_{+}\mid {\psi }_P\left(t\right)\le s\}$, for any $s\in \mathbb{R}$. For convenience, let ${\phi }_P\left(s\right):=s+{\psi }_P^{-1}\left(2s\right)$.

By virtue of Theorem 1, we can finally deduce the quantitative stability results with respect to the optimal value function and the optimal solution set of the optimization model (2) when the probability measure is perturbed.

Theorem 2.

Assume that the conditions in Proposition 1 hold. Let X be a bounded subset of ${\mathbb{R}}^n$ and f be Lipschitz continuous on X with constant $L>0$. Then, we have the following inequalities:

$$|\vartheta \left(Q\right)-\vartheta \left(Q^{\prime }\right)|\le \frac{L}{\tau }\mathcal{D}(Q,Q^{\prime }),\forall Q,Q^{\prime }\in \mathcal{P}$$

(17)

and

$$\mathbb{D}(S\left(Q\right),S\left(P\right))\le {\phi }_P\left(\frac{L}{\tau }\mathcal{D}(Q,P)\right),\forall Q\in \mathcal{P}.$$

(18)

Proof. 

According to the assumptions, it follows from Proposition 1 that, for any $Q\in \mathcal{P}$, $\mathbb{S}\left(Q\right)$ is a closed subset of X. Then, $\mathbb{S}\left(Q\right)$ is compact. It follows from the continuity of f that $S\left(Q\right)\ne \varnothing$. To show the first inequality, we pick $Q,Q^{\prime }\in \mathcal{P},x\in S\left(Q\right)$ and $x^{\prime }\in S\left(Q^{\prime }\right)\subset \mathbb{S}\left(Q^{\prime }\right)$. It follows from Theorem 1 that we can choose $\widehat{x}\in \mathbb{S}\left(Q\right)$ such that

$$\parallel \widehat{x}-x^{\prime }\parallel \le \frac{1}{\tau }\mathcal{D}(Q,Q^{\prime }).$$

(19)

Then, we have the following estimation:

$$\begin{array}{cc}\hfill \vartheta \left(Q\right)-\vartheta \left(Q^{\prime }\right)& =f\left(x\right)-f\left(x^{\prime }\right)=f\left(x\right)-f\left(\widehat{x}\right)+f\left(\widehat{x}\right)-f\left(x^{\prime }\right)\hfill \\ & \le f\left(\widehat{x}\right)-f\left(x^{\prime }\right)\le L\parallel \widehat{x}-x^{\prime }\parallel \le \frac{L}{\tau }\mathcal{D}(Q,Q^{\prime }).\hfill \end{array}$$

Since Q and $Q^{\prime }$ are arbitarily chosen and the general metric $\mathcal{D}(Q,Q^{\prime })$ is symmetric, we also have $\vartheta \left(Q^{\prime }\right)-\vartheta \left(Q\right)\le \frac{L}{\tau }\mathcal{D}(Q,Q^{\prime })$. Therefore, (17) holds.

For the second inequality, we pick arbitary $Q\in \mathcal{P}$ and $x\in S\left(Q\right)$. using Theorem 1, we are able to select $\widehat{x}\in \mathbb{S}\left(P\right)$ such that $\parallel \widehat{x}-x\parallel \le \frac{1}{\tau }\mathcal{D}(Q,P)$. Then, we have

$$\begin{array}{cc}\hfill \frac{2L}{\tau }\mathcal{D}(Q,P)& \ge f\left(\widehat{x}\right)-f\left(x\right)+\frac{L}{\tau }\mathcal{D}(Q,P)\hfill \\ \hfill & \ge f\left(\widehat{x}\right)-f\left(x\right)+\vartheta \left(Q\right)-\vartheta \left(P\right)\hfill \\ & =f\left(\widehat{x}\right)-\vartheta \left(P\right)\ge {\psi }_P\left(d(\widehat{x},S\left(P\right))\right).\hfill \end{array}$$

Therefore, $d(\widehat{x},S\left(P\right))\le {\psi }_P^{-1}\left(\frac{2L}{\tau }\mathcal{D}(Q,P)\right)$. By the triangle inequality, we arrive at

$$\begin{array}{cc}\hfill d(x,S(P\left)\right)& \le \parallel \widehat{x}-x\parallel +d(\widehat{x},S\left(P\right))\hfill \\ & \le \frac{L}{\tau }\mathcal{D}(Q,P)+{\psi }_P^{-1}\left(\frac{2L}{\tau }\mathcal{D}(Q,P)\right)\hfill \\ & ={\phi }_P\left(\frac{L}{\tau }\mathcal{D}(Q,P)\right).\hfill \end{array}$$

Since x is arbitarily chosen from $S\left(Q\right)$, we conclude that inequality (18) holds. □

Remark 1.

In Theorems 1 and 2, the stability results for the optimization model (2) are obtained under the primal condition (8) which ensures the uniform error bound property of the constraint system (6). It is a supplement to the Slater condition which was imposed for the stability analysis of the distributionally robust model obtained in reference [2] (Theorems 4.3 and 4.4).

3.2. Stability Analysis of OPSGE (3)

In this section, we consider optimization model (3) with a set-valued schochastic generalized equation constraint of the following form:

$$0\in {\mathbb{E}}_Q[\Gamma (x,\xi )]+\mathcal{G}\left(x\right),$$

(20)

where $Q\in \mathcal{P}$ is a perturbation of the probability distribution P, $\Gamma :\mathcal{X}\times \mathsf{\Xi }\rightrightarrows \mathcal{Y}$ and $\mathcal{G}:\mathcal{X}\rightrightarrows \mathcal{Y}$ are closed set-valued mappings, $\mathcal{X}\subset {\mathbb{R}}^n$, $\mathcal{Y}\subset {\mathbb{R}}^d$, $\xi :\mathsf{\Omega }\to \mathsf{\Xi }$ is a random vector defined on a probability space $(\mathsf{\Omega },\mathcal{F},P)$ with support set $\mathsf{\Xi }\subset {\mathbb{R}}^m$ and probability distribution P.

For convenience, we denote by $\tilde{\mathbb{S}}\left(Q\right)$ the feasible solution set of (3), i.e.,

$$\tilde{\mathbb{S}}\left(Q\right):=\{x\in \mathcal{X}\mid 0\in {\mathbb{E}}_Q[\Gamma (x,\xi )]+\mathcal{G}\left(x\right)\}.$$

(21)

Furthermore, let $\tilde{S}\left(Q\right)$ and $\tilde{\vartheta }\left(Q\right)$ denote the optimal solution set and the optimal value of model (3), respectively.

To establish quantitative stability results for OPSGE (3) as the probability distribution perturbs, we need to define the following distance for probability measures induced by $\mathcal{F}$:

$$\tilde{\mathcal{D}}(Q,P):=\underset{g\in \mathcal{F}}{sup}({\mathbb{E}}_Q\left[g\left(\xi \right)\right]-{\mathbb{E}}_P\left[g\left(\xi \right)\right]),\forall P,Q\in \mathcal{P},$$

(22)

where $\mathcal{F}$ consists of all functions generated by the support function $\sigma (\Gamma (x,·),u)$ over the set $\mathcal{X}\times B_{{\mathbb{R}}^n}$, i.e.,

$$\mathcal{F}:=\left\{g\right(·)\mid g(\xi )=\sigma (\Gamma (x,\xi ),u)\mathrm{for}x\in \mathcal{X},\parallel u\parallel \le 1\}.$$

It is well acknowledged that $\tilde{\mathcal{D}}$ is not a metric unless the set $\mathcal{F}$ is enriched. Furthermore, $\tilde{\mathcal{D}}$ is not symmetric (for more details, please refer to reference [28] and the references therein).

Theorem 3.

Let $\mathcal{P}\subset \mathcal{P}\left(\mathsf{\Omega }\right)$, $\Gamma :\mathcal{X}\times \mathsf{\Xi }\rightrightarrows \mathcal{Y}$, $\mathcal{G}:\mathcal{X}\rightrightarrows \mathcal{Y}$ be defined as in (20) and $\tilde{\mathbb{S}}$ be defined as in (21) with $\mathrm{dom}\left(\tilde{\mathbb{S}}\right)=\mathcal{P}$. Assume that $\kappa ,\iota \in (0,+\infty )$ satisfies $\kappa \iota <1$. Suppose that the following conditions hold:

(i) 

$\mathcal{X}$ and $\mathcal{Y}$ are compact subsets of ${\mathbb{R}}^n$ and ${\mathbb{R}}^d$, respectively;

(ii) 

Γ takes convex set values in $\mathcal{Y}$ and is upper semicontinuous with respect to x for every $\xi \in \mathsf{\Xi }$. Furthermore, Γ is bounded by a P-integrable function $\rho \left(\xi \right)$ for $x\in \mathcal{X}$;

(iii) 

${\mathsf{\Psi }}_P\left(x\right)={\mathbb{E}}_P[\Gamma (x,\xi )]$ is metrically regular on $\mathcal{X}\times \mathcal{Y}$ with constant κ;

(iv) 

$\mathcal{G}$ is Lipschitz continuous on X with constant ι.

Then, the feasible solution set is closed and we have the following estimate for the variation of the feasible sets with constant $\frac{\kappa }{1-\kappa \iota }$:

$$\mathbb{D}(\tilde{\mathbb{S}}\left(Q\right),\tilde{\mathbb{S}}\left(P\right))\le \frac{\kappa }{1-\kappa \iota }\tilde{\mathcal{D}}(Q,P),\forall Q\in \mathcal{P}.$$

(23)

Proof. 

For any $x\in \mathcal{X}$ and any sequence $\left\{x_k\right\}\subset \mathcal{X}$ with $x_k\to x$, since $\Gamma$ is closed, upper semicontinuous and integrably bounded, it follows from reference [29] (Theorem 2.8) that

$$\underset{k\to \infty }{lim\; sup}{\mathsf{\Psi }}_P\left(x_k\right)=\underset{k\to \infty }{lim\; sup}{\mathbb{E}}_P[\Gamma (x_k,\xi )]\subset {\mathbb{E}}_P[\underset{k\to \infty }{lim\; sup}\Gamma (x_k,\xi )]\subset {\mathbb{E}}_P[\Gamma (x,\xi )]={\mathsf{\Psi }}_P\left(x\right).$$

Therefore, the mapping ${\mathsf{\Psi }}_P$ is closed. Pick any $Q\in \mathcal{P}$ and any sequence $\left\{x_k\right\}\subset \tilde{\mathbb{S}}\left(Q\right)$ with $x_k\to x$, we have $0\in {\mathsf{\Psi }}_P\left(x_k\right)+\mathcal{G}\left(x_k\right)$, then there exists $y_k\in \mathcal{Y}$ such that $y_k\in {\mathsf{\Psi }}_P\left(x_k\right)$ and $-y_k\in \mathcal{G}\left(x_k\right)$ (for each $k\in \mathbb{N}$). Since $\mathcal{Y}$ is compact, without loss of generality, we may assume that $y_k\to y\in \mathcal{Y}$. Through this and the closedness of ${\mathsf{\Psi }}_P$ and $\mathcal{G}$, one has $y\in {\mathsf{\Psi }}_P\left(x\right)$ and $-y\in \mathcal{G}\left(x\right)$. Then, $0\in {\mathsf{\Psi }}_P\left(x\right)+\mathcal{G}\left(x\right)$, i.e., $x\in \tilde{\mathbb{S}}\left(Q\right)$. Hence, the feasible solution set is closed.

From assumptions (i) and (ii), it is easy to observe that ${\mathsf{\Psi }}_P$ takes convex and compact set values in $\mathcal{Y}$. Then, it follows from Lemma 2.5 that, for all $x\in \mathcal{X}$ and $Q\in \mathcal{P}$,

$$\mathbb{D}({\mathsf{\Psi }}_Q\left(x\right),{\mathsf{\Psi }}_P\left(x\right))=\underset{\parallel u\parallel \le 1}{max}(\sigma ({\mathbb{E}}_Q[\Gamma (x,\xi )],u)-\sigma \left({\mathbb{E}}_P[(\Gamma (x,\xi )],u)\right).$$

According to reference [30], Proposition 3.4, we have

$${\mathbb{E}}_Q\left[\sigma (\Gamma (x,\xi ),u)\right]-{\mathbb{E}}_P\left[\sigma (\Gamma (x,\xi ),u)\right]=\sigma ({\mathbb{E}}_Q[\Gamma (x,\xi )],u)-\sigma ({\mathbb{E}}_P[(\Gamma (x,\xi )],u).$$

Then, it follows from the definition of $\tilde{\mathcal{D}}(Q,P)$ that

$$\mathbb{D}({\mathsf{\Psi }}_Q\left(x\right),{\mathsf{\Psi }}_P\left(x\right))\le \tilde{\mathcal{D}}(Q,P),\forall x\in \mathcal{X},Q\in \mathcal{P}.$$

(24)

By assumptions (iii) and (iv), one has

$$d(x,{\mathsf{\Psi }}_P^{-1}\left(y\right))\le \kappa d(y,{\mathsf{\Psi }}_P\left(x\right)),\forall (x,y)\in \mathcal{X}\times \mathcal{Y}$$

(25)

and

$$\mathbb{H}(\mathcal{G}\left(x\right),\mathcal{G}\left(x^{\prime }\right))\le \iota \parallel x^{\prime }-x\parallel ,\forall x^{\prime },x\in \mathcal{X}.$$

(26)

Since $\mathrm{dom}\left(\tilde{\mathbb{S}}\right)=\mathcal{P}$, it follows from (25) that

$${\mathsf{\Psi }}_P^{-1}\left(y\right)\ne \varnothing ,\forall y\in \mathcal{Y}$$

(27)

and from (26) that

$$\mathcal{G}\left(x\right)\ne \varnothing ,\forall x\in \mathcal{X}.$$

(28)

Let $\epsilon \in (0,1)$ be sufficient small such that

$$\begin{array}{cc}\hfill & \kappa (\iota +\epsilon )+\epsilon <1.\hfill \end{array}$$

(29)

To show that (23) holds, we pick any $Q\in \mathcal{P}$ with $Q\ne P$ and let it be fixed, then it suffices to show that

$$\tilde{\mathbb{S}}\left(Q\right)\subset \tilde{\mathbb{S}}\left(P\right)+\frac{\kappa }{1-\kappa \iota }\tilde{\mathcal{D}}(Q,P)B_{R^n}.$$

(30)

Let $x_0\in \tilde{\mathbb{S}}\left(Q\right)$, i.e., $x_0\in \mathcal{X}$ and $0\in {\mathsf{\Psi }}_Q\left(x_0\right)+\mathcal{G}\left(x_0\right)$. Then, there exists $y_0\in \mathcal{Y}$ such that $y_0\in {\mathsf{\Psi }}_Q\left(x_0\right)$ and $-y_0\in \mathcal{G}\left(x_0\right)$. It follows from (24), (25) and (27) that, there exists $x_1\in {\mathsf{\Psi }}_P^{-1}\left(y_0\right)$ such that

$$\begin{array}{cc}\hfill \parallel x_1-x_0\parallel & <d(x_0,{\mathsf{\Psi }}_P^{-1}\left(y_0\right))+\epsilon \le \kappa d(y_0,{\mathsf{\Psi }}_P\left(x_0\right))+\epsilon \hfill \\ \hfill & \le \kappa (\mathbb{D}({\mathsf{\Psi }}_Q\left(x_0\right),{\mathsf{\Psi }}_P\left(x_0\right))+\epsilon \hfill \\ \hfill & \le \kappa \tilde{\mathcal{D}}(Q,P)+\epsilon .\hfill \end{array}$$

(31)

If $x_1=x_0$, then $y_0\in {\mathsf{\Psi }}_Q\left(x_0\right)$, and then $x_0\in \tilde{\mathbb{S}}\left(Q\right)$. This shows that (30) holds automatically. Let $x_1\ne x_0$, then $\parallel x_1-x_0\parallel >0$. According to (26) and (28), there exists $-y_1\in \mathcal{G}\left(x_1\right)$ such that

$$\begin{array}{cc}\hfill \parallel y_1-y_0\parallel & <d(-y_0,\mathcal{G}\left(x_1\right))+\epsilon \parallel x_1-x_0\parallel \hfill \\ \hfill & \le \mathbb{H}(\mathcal{G}\left(x_0\right),\mathcal{G}\left(x_1\right))+\epsilon \parallel x_1-x_0\parallel \hfill \\ \hfill & \le (\iota +\epsilon )\parallel x_1-x_0\parallel .\hfill \end{array}$$

(32)

By induction, we construct sequences of points $x_k\in \mathcal{X}$ and $y_k\in \mathcal{Y}$ such that, for $k=0,1,2,\dots$,

$$x_{k+1}\in {\mathsf{\Psi }}_P^{-1}\left(y_k\right)\mathrm{and}-y_{k+1}\in \mathcal{G}\left(x_{k+1}\right)$$

(33)

with

$$\parallel x_{k+1}-x_k{\parallel \le (\kappa (\iota +\epsilon )+\epsilon )}^k\parallel x_1-x_0\parallel \mathrm{and}\parallel y_{k+1}-y_k\parallel \le (\iota +\epsilon )\parallel x_{k+1}-x_k\parallel .$$

(34)

According to (32), we see that $x_1$ and $y_1$ satisfy (33) and (34) with $k=0$. Suppose that for some $n\ge 1$, we have generated $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ such that (33) and (34) hold. Our goal is to show that there exist $x_{n+1}\in \mathcal{X}$ and $y_{n+1}\in \mathcal{Y}$ which satisfy (33) and (34). If $x_n=x_{n-1}$, we set $x_{n+1}=x_n$ and $y_{n+1}=y_n$. Otherwise, since $x_n\in {\mathsf{\Psi }}_P^{-1}\left(y_{n-1}\right)$, it follows from (25) and (27) that there exists $x_{n+1}\in {\mathsf{\Psi }}_P^{-1}\left(y_n\right)$ such that

$$\begin{array}{cc}\hfill \parallel x_{n+1}-x_n\parallel & \le d(x_n,{\mathsf{\Psi }}_P^{-1}\left(y_n\right))+\epsilon \parallel x_n-x_{n-1}\parallel \hfill \\ \hfill & \le \kappa d(y_n,{\mathsf{\Psi }}_P\left(x_n\right))+\epsilon \parallel x_n-x_{n-1}\parallel \hfill \\ & \le \kappa \parallel y_n-y_{n-1}\parallel +\epsilon \parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

Then, by invoking the induction hypothesis (34), we have $\parallel y_n-y_{n-1}\parallel \le (\iota +\epsilon )\parallel x_n-x_{n-1}\parallel$. And then,

$$\parallel x_{n+1}-x_n\parallel \le (\kappa (\iota +\epsilon )+\epsilon )\parallel x_n-x_{n-1}{\parallel \le (\kappa (\iota +\epsilon )+\epsilon )}^n\parallel x_1-x_0\parallel .$$

If $x_{n+1}=x_n$, we set $y_{n+1}=y_n$. If $x_{n+1}\ne x_n$, note that $-y_n\in \mathcal{G}\left(x_n\right)$, according to (28) there exists $-y_{n+1}\in \mathcal{G}\left(x_{n+1}\right)$ such that

$$\parallel y_{n+1}-y_n\parallel \le \mathbb{H}(\mathcal{G}\left(x_{n+1}\right),\mathcal{G}\left(x_n\right))+\epsilon \parallel x_{n+1}-x_n\parallel \le (\iota +\epsilon )\parallel x_{n+1}-x_n\parallel .$$

This completes the induction step, and hence (33) and (34) hold true for all $k\in \mathbb{N}$.

By virtue of the inequalities for $x_k$ and $y_k$ in (34), we observe that for any positive integers m and n, we have

$$\begin{array}{cc}\hfill \parallel x_{n+m}-x_n\parallel & \le \sum _{k=n}^{n+m-1}\parallel x_{k+1}-x_k\parallel \le \sum _{k=n}^{n+m-1}{(\kappa (\iota +\epsilon )+\epsilon )}^k\parallel x_1-x_0\parallel \hfill \\ \hfill & \le \frac{{(\kappa (\iota +\epsilon )+\epsilon )}^n}{1-\left(\kappa \right(\iota +\epsilon )+\epsilon )}\parallel x_1-x_0\parallel \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel y_{n+m}-y_n\parallel & \le \sum _{k=n}^{n+m-1}\parallel y_{k+1}-y_k\parallel \le \sum _{k=n}^{n+m-1}(\iota +\epsilon )\parallel x_{k+1}-x_k\parallel \hfill \\ \hfill & \le \frac{(\mu +\epsilon ){(\kappa (\iota +\epsilon )+\epsilon )}^n}{1-\left(\kappa \right(\iota +\epsilon )+\epsilon )}\parallel x_1-x_0\parallel ,\hfill \end{array}$$

which indicates that both sequences $\left\{x_k\right\}$ and $\left\{y_k\right\}$ are Cauchy sequences. Note that $(x_k,y_{k-1})\in \mathrm{gph}\left({\mathsf{\Psi }}_P\right)$ and $(x_k,-y_k)\in \mathrm{gph}\left(\mathcal{G}\right)$, since $\mathrm{gph}\left({\mathsf{\Psi }}_P\right)$ and $\mathrm{gph}\left(\mathcal{G}\right)$ are closed, then there exists $(x,y)\in \mathrm{gph}\left({\mathsf{\Psi }}_P\right)$ such that $(x_k,y_{k-1})\to (x,y)$, and then $(x,-y)\in \mathrm{gph}\left(\mathcal{G}\right)$. This shows that $x\in \tilde{\mathbb{S}}\left(P\right)$. Utilizing (31) and (34), we finally obtain that

$$\begin{array}{cc}\hfill d(x_0,\tilde{\mathbb{S}}\left(P\right))& \le \parallel x_0-x\parallel =\underset{k\to \infty }{lim}\parallel x_k-x_0\parallel \le \sum _{k=0}^{\infty }\parallel x_{k+1}-x_k\parallel \hfill \\ \hfill & \le \frac{1}{1-\left(\kappa \right(\iota +\epsilon )+\epsilon )}\parallel x_1-x_0\parallel \hfill \\ \hfill & \le \frac{\kappa \tilde{\mathcal{D}}(Q,P)+\epsilon }{1-\left(\kappa \right(\iota +\epsilon )+\epsilon )}.\hfill \end{array}$$

Taking the limit as $\epsilon \downarrow 0$ gives us (30), which completes the proof. □

Remark 2.

In reference [28], the authors provided qualitative stability analysis for feasible solution mapping under a metric regularity assumption on the set-valued mapping ${\mathbb{E}}_P[\Gamma (·,\xi )]+\mathcal{G}(·)$. In contrast, Theorem 3 imposes metric regularity on the mapping ${\mathbb{E}}_P[\Gamma (·,\xi )]$ and Lipschitz continuity on $\mathcal{G}$, which does not necessarily guarantee the metric regularity of the mapping ${\mathbb{E}}_P[\Gamma (·,\xi )]+\mathcal{G}(·)$. Furthermore, the proof of Theorem 3 is completely self-contained. Hence, Theorem 3 is a supplement to the results in reference [28] (Theorem 3.1 (iii)). In reference [31], the authors focus on the case when the mapping Γ is single-valued and obtained similar stability results. Therefore, Theorem 3 also extends the results in reference [31] (Theorem 8).

With the help of Theorem 3, we are able to deduce the following quantitative stability results regarding the optimal value function.

Theorem 4.

Under the assumptions of Theorem 3, assume that f is Lipschitz continuous on $\mathcal{X}$ with constant $L>0$. Then, the optimal solution mapping $\tilde{S}$ of model (3) is compact valued, $\mathrm{dom}\left(\tilde{S}\right)=\mathcal{P}$, and

$$\tilde{\vartheta }\left(P\right)\le \tilde{\vartheta }\left(Q\right)+\frac{\kappa L}{1-\kappa \iota }\tilde{\mathcal{D}}(Q,P),\forall Q\in \mathcal{P}.$$

(35)

Proof. 

It is shown in Theorem 3 that the feasible solution set of model (3) is closed. Since $\mathcal{X}$ is compact, we know that $\tilde{\mathbb{S}}\left(Q\right)$ is compact for all $Q\in \mathcal{P}$. Due to the continuity of f, we have $\tilde{S}\left(Q\right)\ne \varnothing$, for all $Q\in \mathcal{P}$, hence $\mathrm{dom}\left(\tilde{S}\right)=\mathcal{P}$. Let $Q\in \mathcal{P}$ and $\left\{x_k\right\}\subset \tilde{S}\left(Q\right)$ with $x_k\to x$, one has

$$\tilde{\vartheta }\left(Q\right)=f\left(x_k\right)=\underset{k\to \infty }{lim}f\left(x_k\right)=f\left(x\right),$$

which indicates that $x\in \tilde{S}\left(Q\right)$. Taking into account of the fact that $\mathcal{X}$ is compact, we ascertain that $\tilde{S}$ is compact valued.

To establish (35), we pick $Q\in \mathcal{P},x\in \tilde{S}\left(Q\right)$ and $x^{\prime }\in \tilde{S}\left(P\right)$. Note that $x\in \tilde{S}\left(Q\right)\subset \tilde{\mathbb{S}}\left(Q\right)$; it follows from Theorem 3 that, for any $\epsilon >0$, we can choose $\widehat{x}\in \tilde{\mathbb{S}}\left(P\right)$ such that

$$\parallel \widehat{x}-x\parallel \le \frac{\kappa }{1-\kappa \iota }\tilde{\mathcal{D}}(Q,P)+\epsilon .$$

Then, we have the following estimation:

$$\begin{array}{cc}\hfill \tilde{\vartheta }\left(P\right)-\tilde{\vartheta }\left(Q\right)& =f\left(x^{\prime }\right)-f\left(x\right)=f\left(x^{\prime }\right)-f\left(\widehat{x}\right)+f\left(\widehat{x}\right)-f\left(x\right)\hfill \\ & \le f\left(\widehat{x}\right)-f\left(x\right)\le L\parallel \widehat{x}-x\parallel \le \frac{\kappa L}{1-\kappa \iota }\tilde{\mathcal{D}}(Q,P)+L\epsilon .\hfill \end{array}$$

By letting $\epsilon \to 0$, we arrive at (35), which completes the proof. □

4. Conclusions

The main contribution of this paper contains two parts. Firstly, we establish a new primal sufficient condition for the uniform error bound property of a constraint system (6) by utilizing the Ekeland’s Variational Principle. Based on that, we then study the quantitative stability for OPSC (2) and OPSGE (3), respectively, by estimating the variation of the optimal value function, the optimal solution set and the feasible solution set of the aforementioned optimization models when the underlying probability distribution perturbs. The obtained results are new, and they are supplementary to and extensions of the existing ones in the literature. The proof is completely self-contained.