Quantitative stability analysis of stochastic quasi-variational inequality problems and applications

Abstract

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints.

Appendix

Proof of Lemma 2.2

Part (i). Observe that (2.4) implies that $\Psi (\bar{u}) \ne \emptyset$, and under uniform continuity w.r.t. u further that $\Psi (u) \ne \emptyset$ when u is sufficiently close to $\bar{u}$.

It is well known (see [3, Definition 1.4.6]) that

$$\begin{aligned} \liminf _{u\rightarrow \bar{u}}\Psi (u)\subseteq {\mathrm{cl}} \; \Psi (\bar{u})\subseteq \limsup _{u\rightarrow \bar{u}}\Psi (u). \end{aligned}$$

Since $\Psi (\bar{u})$ is closed, to show continuity of $\Psi (\cdot )$ at $\bar{u}$, it suffices to show that the set-valued mapping is both upper and lower semicontinuous at $\bar{u}$, that is,

$$\begin{aligned} \limsup _{u\rightarrow \bar{u}}\Psi (u)\subseteq \Psi (\bar{u})\subseteq \liminf _{u\rightarrow \bar{u}}\Psi (u). \end{aligned}$$

(6.1)

The upper semicontinuity can be easily verified under the continuity of $\psi$ and closeness of $\mathcal{K}$. Hence it suffices to show the lower semicontinuity.

Observe that condition (2.4) and continuity of $\psi (\cdot ,\bar{u})$ imply that ${\mathrm{int}} \; \Psi (\bar{u})\ne \emptyset$. Moreover, since $\psi (z,u)$ is uniformly continuous w.r.t. u under condition (b), it is easy to show that for any $z \in {\mathrm{int}} \; \Psi (\bar{u})$, we can set u sufficiently close to $\bar{u}$ such that $z\in \Psi (u)$. This allows us to claim that

$$\begin{aligned} {\mathrm{int}} \; \Psi (\bar{u}) \subseteq \liminf _{u\rightarrow \bar{u}} \Psi (u). \end{aligned}$$

(6.2)

Moreover, since $\Psi (\bar{u})$ is convex and closed and by Rockafellar and Wets [26, Proposition 4.4], $\liminf _{u\rightarrow \bar{u}} \Psi (u)$ is closed, we have

$$\begin{aligned} \Psi (\bar{u}) = {\mathrm{cl}}\; \left( {\mathrm{int}} \; \Psi (\bar{u})\right) \subseteq {\mathrm{cl}}\;(\liminf _{u\rightarrow \bar{u}} \Psi (u)) = \liminf _{u\rightarrow \bar{u}} \Psi (u), \end{aligned}$$

which yields the second inclusion of (6.1).

Part (ii). Since $\Psi (\bar{u})$ is compact and $\Psi (\cdot )$ is closed at $\bar{u}$, the continuity of $\Psi (\cdot )$ at $\bar{u}$ means that for any number $\delta >0$, there exists $\epsilon >0$ such that when $\Vert u-\bar{u}\Vert _U\le \epsilon$,

$$\begin{aligned} \mathbb {H} (\Psi (u),\Psi (\bar{u})) \le \delta \end{aligned}$$

or equivalently

$$\begin{aligned} \Psi (u)\subseteq \Psi (\bar{u}) + \delta \mathcal{B}_m \; {\mathrm{and}} \; \Psi (\bar{u}) \subseteq \Psi (u) + \delta \mathcal{B}_m. \end{aligned}$$

(6.3)

In what follows, we use Lemma 2.1 (Robinson–Ursescu’s theorem) to derive an error bound for set-valued mapping $\Psi (\cdot )$ at $\bar{u}$ under condition (d) and Slater constraint qualification (2.4).

Let

$$\begin{aligned} \mathcal {F}_{u}(z):=\left\{ \begin{array}{cl} \psi (z,u)-\mathcal{K}, &{}\quad \text{ for }\,\, z\in \mathcal{Z}, \\ \emptyset , &{} \quad \text{ for }\,\, z\notin \mathcal{Z}. \end{array} \right. \end{aligned}$$

Condition (a) ensures that $\mathcal {F}_{u}(z)$ is closed and convex set-valued over its domain $\mathcal{Z}$. Moreover,

$$\begin{aligned} \Psi (u) = \mathcal {F}_{u}^{-1}(0), \end{aligned}$$

(6.4)

and $z\in \Psi (u)$ if and only if $(z,0)\in \text{ gph }\; \mathcal {F}_{u}$. Furthermore, it follows from (2.4)

$$\begin{aligned} 0\in \text{ int }(\text{ range } \; \mathcal {F}_{\bar{u}}). \end{aligned}$$

(6.5)

Let $z \in \Psi (\bar{u})$ be fixed, that is, $(z,0)\in \text{ gph }\; \mathcal {F}_{\bar{u}}$. By Lemma 2.1, there exist positive numbers $\delta _{z}$, $\eta _z$ and $c_{z}$ (depending on z) such that

$$\begin{aligned} d(z', \mathcal{F}_{\bar{u}}^{-1}(q)) \le c_{z} d(q,\mathcal{F}_{\bar{u}}(z')). \end{aligned}$$

(6.6)

for each $z'\in B_m(z,\delta _{z})\cap \mathcal{Z}$ and $q\in B_W(0,\eta _z)$. In particular,

$$\begin{aligned} d(z', \Psi (\bar{u})) = d(z',\mathcal {F}_{\bar{u}}^{-1}(0)) \le c_{z} d(\psi (z',\bar{u}), \mathcal{K}). \end{aligned}$$

(6.7)

Under condition (d), we have

$$\begin{aligned} d(\psi (z',\bar{u}), \mathcal{K}) \le \Vert \psi (z', \bar{u})-\psi (z', u)\Vert _Z +d(\psi (z',u), \mathcal{K}) \le \sigma \Vert u-\bar{u}\Vert _U^\nu . \end{aligned}$$

(6.8)

for $z'\in \Psi (u)$.

On the other hand, since z is arbitrarily chosen from $\Psi (\bar{u})$, the set $\Psi (\bar{u})$ may be covered by the union of a collection of $\delta$-balls, i.e.,

$$\begin{aligned} \Psi (\bar{u})\subseteq \bigcup _{z\in \Psi (\bar{u})}\text{ int }\,\, B_m(z,\delta _{z}). \end{aligned}$$

Since $\Psi (\bar{u})$ is a compact set, by the finite covering theorem, there exist a finite number of points $z_1, z_2,\dots , z_k \in \Psi (\bar{u})$ and positive constants $\delta _{z_i}, c_{z_i}, i=1,\dots ,k$ such that

$$\begin{aligned} \Psi (\bar{u})\subseteq \bigcup _{i=1}^k\text{ int }\,\,B_m(z_i,\delta _{z_i}) \end{aligned}$$

(6.9)

and

$$\begin{aligned} d(z', \Psi (\bar{u}))\le c_{z_i} d(\psi (z',\bar{u}), \mathcal{K}) \end{aligned}$$

(6.10)

for any $z'\in B_m(z_i,\delta _{z_i})\cap \mathcal{Z}$. Since $\Psi (\bar{u})$ is a compact set and $\bigcup _{i=1}^k\text{ int }\,\,B_m(z_i,\delta _{z_i})$ is an open set, there exists a positive constant $\delta$ such that

$$\begin{aligned} \Psi (\bar{u})+\delta \mathcal{B}_m\subseteq \bigcup _{i=1}^k\text{ int }\,\,B_m(z_i,\delta _{z_i}). \end{aligned}$$

(6.11)

Let $C:=\max \{c_{z_1},\dots ,c_{z_k}\}$. Combining (6.11) and (6.10), we obtain

$$\begin{aligned} d(z', \Psi (\bar{u}))\le Cd(\psi (z',\bar{u}), \mathcal{K}) \end{aligned}$$

(6.12)

for all $z'\in \left( \Psi (\bar{u})+\delta \mathcal{B}_m\right) \cap \mathcal{Z}$. Through (6.3) and (6.8), we arrive at

$$\begin{aligned} \mathbb {D}(\Psi (u), \Psi (\bar{u}))\le Cd(\psi (z,\bar{u}), \mathcal{K}) \le C\sigma \Vert u-\bar{u}\Vert _U^\nu \end{aligned}$$

for all $z'\in \Psi (u)$.

To complete the proof of (2.5), it suffices to show

$$\begin{aligned} \mathbb {D}(\Psi (\bar{u}),\Psi (u)) \le C\sigma \Vert u-\bar{u}\Vert _U^\nu . \end{aligned}$$

(6.13)

For any $z\in \mathcal{Z}$, let $q=\psi (z,\bar{u})-\psi (z,u)$. Under condition (d), $\Vert q\Vert _W\le \sigma \Vert u-\bar{u}\Vert _U^\nu .$ Moreover, it is easy to see that $\Psi (u) =\mathcal{F}^{-1}_{\bar{u}}(q).$ Let $z\in \Psi (\bar{u})$, $\delta _z$ be defined as in (6.6) and $\epsilon$ be defined at the beginning of the proof of this part with $\sigma \epsilon ^\nu \le \delta _z.$ Under condition (d), we have $\Vert q\Vert \le \delta _z$. It follows from (6.6) (by setting $z'=z$) that

$$\begin{aligned} d(z, \Psi (u))= d(z,\mathcal {F}_{\bar{u}}^{-1}(q)) \le c_{z} d(q,\mathcal{F}_{\bar{u}}(z)). \end{aligned}$$

(6.14)

Since $z\in \Psi (\bar{u})$, then $0\in \mathcal{F}_{\bar{u}}(z)$ and hence

$$\begin{aligned} d(q,\mathcal{F}_{\bar{u}}(z)) \le \Vert q\Vert _W\le \sigma \Vert u-\bar{u}\Vert _U^\nu . \end{aligned}$$

Combining the two inequalities above, we obtain

$$\begin{aligned} d(z,\Psi (u)) \le c_z\sigma \Vert u-\bar{u}\Vert _U^\nu \end{aligned}$$

for any $z\in \Psi (\bar{u})$ and $\Vert u-\bar{u}\Vert _U\le \epsilon$. Utilizing the second inclusion in (6.3), we can apply the finite covering theorem to set $\Psi (\bar{u})$ (similar to previous discussion) and find a positive constant C such that (6.13) holds. The proof is complete. $\square$

In the case when $\psi$ is continuously differentiable, Lemma 2.2 coincides with [28, Corollary 1]. Thus we may regard our result as an extension of the Robinson’s result.

Proof of Lemma 2.3

Let $\hat{\phi }(z,u):=\phi (z,u)+\delta _{\Psi (u)}(z)$, where $\delta _C(\cdot )$ denotes the indicator function of a set $C\subseteq \mathbb {R}^m$ with $\delta _C(x)=0$ for $x\in C$ and $\delta _C(x)=+\infty$ for $x\notin C$. Problem (2.3) can be equivalently written as

$$\begin{aligned} \min _{z\in \mathbb {R}^m}\hat{\phi }(z,u). \end{aligned}$$

The objective function $\hat{\phi }$ has the following properties:

(i):

Since $\phi$ and $\psi$ are continuous, $\mathcal{K}$ and $\mathcal{Z}$ are closed and convex, we have graph$\Psi$ is closed, which means that $\mathrm{epi}\,\delta _{\Psi (\cdot )}(\cdot )=\{(u,z, \alpha ): z\in \Psi (u), \alpha \ge 0\}=\mathrm{graph} \Psi \times \mathbb {R}_+$ is closed. Hence $\hat{\phi }$ is proper lower semicontinuous;

(ii):

Since $\mathcal{Z}$ is compact, $\hat{\phi }$ is level-bounded in z locally uniformly in u;

(iii):

Under the Slater condition (2.4), one has from Lemma 2.2 that

$$\begin{aligned} \lim _{u\rightarrow \bar{u}}\Psi (u)=\Psi (\bar{u}), \end{aligned}$$

which means that

$$\begin{aligned}\limsup _{u\rightarrow \bar{u}}\mathrm {epi}\, [\delta _{\Psi (u)}(\cdot )]=\mathrm {epi}\, [\delta _{\Psi (\bar{u})}(\cdot )],\end{aligned}$$

where

$$\begin{aligned}\mathrm {epi}\, [\delta _{\Psi (u)}(\cdot )]=\{(z,\alpha ): z\in \Psi (u), \alpha \ge 0\}=\Psi (u)\times \mathbb {R}_+.\end{aligned}$$

Then we obtain from Definition 2.1 that

$$\begin{aligned}e\!-\!\!\!\lim _{u\rightarrow \bar{u}}\delta _{\Psi (u)}(\cdot )=\delta _{\Psi (\bar{u})}(\cdot ),\end{aligned}$$

which, by Proposition 2.1, means that for any sequence $u^{\nu }\rightarrow \bar{u}$ and point $\bar{z}\in \mathbb {R}^m$,

$$\begin{aligned}\liminf _{\nu \rightarrow \infty }\delta _{\Psi (u^{\nu })}(z^{\nu })\ge \delta _{\Psi (\bar{u})}(\bar{z})\end{aligned}$$

for every sequence $z^{\nu }\rightarrow \bar{z}$ and there exists some $z^{\nu }\rightarrow \bar{z}$ such that

$$\begin{aligned}\limsup _{\nu \rightarrow \infty }\delta _{\Psi (u^{\nu })}(z^{\nu })\le \delta _{\Psi (\bar{u})}(\bar{z}).\end{aligned}$$

We know from the continuity of $\phi$ that

$$\begin{aligned}\lim _{\nu \rightarrow \infty } \phi (z^{\nu }, u^{\nu })=\phi (\bar{z},\bar{u})\,\,\text{ as }\,\, \nu \rightarrow \infty ,\end{aligned}$$

which leads to

$$\begin{aligned}\liminf _{\nu \rightarrow \infty }\,[\phi (z^{\nu }, u^{\nu })+\delta _{\Psi (u^{\nu })}(z^{\nu })]\ge \phi (\bar{z}, \bar{u})+\delta _{\Psi (\bar{u})}(\bar{z})\end{aligned}$$

and there exists $z^{\nu }\rightarrow \bar{z}$ such that

$$\begin{aligned} \limsup _{\nu \rightarrow \infty }\,[\phi (z^{\nu }, u^{\nu })+\delta _{\Psi (u^{\nu })}(z^{\nu })]\le \phi (\bar{z}, \bar{u})+\delta _{\Psi (\bar{u})}(\bar{z}).\end{aligned}$$

Then also by Proposition 2.1, we have that $u\mapsto \hat{\phi }(\cdot ,u)$ is epi-continuous at $\bar{u}$.

Hence all conditions in [26, Theorem 7.41] are satisfied, and from this theorem we obtain that ${\vartheta }(\cdot )$ is continuous at $\bar{u}$ and $\mathcal{Z}(\cdot )$ is outer semicontinuous at $\bar{u}$. Noting that $\mathcal{Z}(u)\subset \mathcal{Z}$ and it actually is locally bounded around $\bar{u}$, we have from [26, Theorem 5.19] that $\mathcal{Z}(\cdot )$ is upper semicontinuous at $\bar{u}$. $\square$

Proof of Theorem 2.1

Part (i). First we show that

$$\begin{aligned} \mathbb {D}\left( \mathcal{Z}(u), \mathcal{Z}(\bar{u})\right) \le \max _{z \in \mathcal{Z}(u)}\Vert z-\Pi _{\Psi (\bar{u})}(z)\Vert + R(u), \end{aligned}$$

(6.15)

where $\Pi _{\Psi (\bar{u})}(z)$ denotes orthogonal projection of z on $\Psi (\bar{u})$,

$$\begin{aligned} R(u) := \left( \frac{1}{\alpha }\left[ 2\varrho \Vert u-\bar{u}\Vert _U^\gamma + L\left( \max _{z \in \mathcal{Z}(u)}\Vert z-\Pi _{\Psi (\bar{u})}(z)\Vert +\min _{z \in \mathcal{Z}(\bar{u})}\Vert z-\Pi _{\Psi (u)}(z)\Vert \right) \right] \right) ^{\frac{1}{2}}.\nonumber \\ \end{aligned}$$

(6.16)

Let $z(u) \in \mathcal{Z}(u)$ and $z(\bar{u}) \in \mathcal{Z}(\bar{u})$. Under conditions (b) and (c)

$$\begin{aligned} {\vartheta }(u)-\phi (\Pi _{\Psi (\bar{u})}(z(u)),\bar{u})= & {} \phi (z(u),u)-\phi (\Pi _{\Psi (\bar{u})}(z(u)),\bar{u}) \nonumber \\\ge & {} -L\Vert z(u) - \Pi _{\Psi (\bar{u})}(z(u))\Vert -\varrho \Vert u-\bar{u}\Vert _U^{\gamma } \quad \end{aligned}$$

(6.17)

for $u\in B(\bar{u},\bar{\delta }_u)$. On the other hand, since $\Pi _{\Psi (u)}(z(\bar{u}))\in \Psi (u)$ and z(u) is an optimal solution to $\mathcal{P}_{u}$, we have

$$\begin{aligned} {\vartheta }(u) \le \phi \left( \Pi _{\Psi (u)}(z(\bar{u})),u\right) . \end{aligned}$$

Using this inequality, (2.8) and the growth condition (c), we have

$$\begin{aligned} {\vartheta }(u) - \phi (\Pi _{\Psi (\bar{u})}(z(u)),\bar{u})= & {} \phi (z(u), u)-\phi (z(\bar{u}),u)+(\phi (z(\bar{u}),u)-\phi (z(\bar{u}),\bar{u}))\nonumber \\&-\,(\phi (\Pi _{\Psi (\bar{u})}(z(u)),\bar{u})-\phi (z(\bar{u}),\bar{u}))\nonumber \\\le & {} \phi (\Pi _{\Psi (u)}(z(\bar{u})),u) -\phi (z(\bar{u}),u) + \varrho \Vert u-\bar{u}\Vert _U^{\gamma }\nonumber \\&-\,\alpha d\left( \Pi _{\Psi (\bar{u})}(z(u)), \mathcal{Z}(\bar{u})\right) ^2\nonumber \\\le & {} L\Vert \Pi _{\Psi (u)}(z(\bar{u}))-z(\bar{u})\Vert +\varrho \Vert u-\bar{u}\Vert _U^{\gamma }\nonumber \\&-\,\alpha d\left( \Pi _{\Psi (\bar{u})}(z(u)), \mathcal{Z}(\bar{u})\right) ^2 \end{aligned}$$

(6.18)

for $u\in B(\bar{u},\bar{\delta }_u)$. Combining (6.17) and (6.18), we obtain

$$\begin{aligned}&d\left( \Pi _{\Psi (\bar{u})}(z(u)), \mathcal{Z}(\bar{u})\right) \nonumber \\&\quad \le \left( [L\Vert z(u)-\Pi _{\Psi (\bar{u})}(z(u))\Vert +2\varrho \Vert u-\bar{u}\Vert _U^{\gamma } +L\Vert z(\bar{u})-\Pi _{\Psi (u)}(z(\bar{u}))\Vert ]/\alpha \right) ^{\frac{1}{2}}, \nonumber \\ \end{aligned}$$

(6.19)

which yields (6.15) through the triangle inequality below

$$\begin{aligned} d\left( z(u), \mathcal{Z}(\bar{u})\right) \le \Vert z(u)-\Pi _{\Psi (\bar{u})}(z(u))\Vert +d\left( \Pi _{\Psi (\bar{u})}(z(u)),\mathcal{Z}(\bar{u})\right) \end{aligned}$$

(6.20)

because z(u) and $z(\bar{u})$ are arbitrarily taken from the set of optimal solutions. Since $\mathcal {Z}(u)\subseteq \Psi (u)$, it follows by Lemma 2.2 that there exists a constant $\epsilon >0$ such that

$$\begin{aligned} \max \left\{ \max _{z\in \mathcal {Z}(u)}\Vert z-\Pi _{\Psi (\bar{u})}(z)\Vert , \min _{z\in \mathcal {Z}(\bar{u}) }\Vert z-\Pi _{\Psi (u)}(z)\Vert \right\} =O(\Vert u-\bar{u}\Vert _U^\nu ) \end{aligned}$$

(6.21)

when $\Vert u-\bar{u}\Vert _U\le \epsilon$. Combining (6.15) and (6.21), we obtain (2.9).

Part (ii). We only need to show that

$$\begin{aligned} \mathbb {D}\left( \mathcal{Z}(\bar{u}), \mathcal{Z}(u)\right) \le c'\Vert u-\bar{u}\Vert _U^{\beta }. \end{aligned}$$

From the proof of Part (i), we can see that we can swap u with $\bar{u}$ except that the first inequality in formulae (6.18) requires second order growth condition of $\phi (\cdot ,u)$ over the optimal solution set $\mathcal{Z}(u)$. $\square$