Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

Abstract

We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA solutions. Our analysis exploits problem structure in PDE-constrained optimization problems, allowing us to construct deterministic, compact subsets of the feasible set that contain the solutions to the risk-neutral problem and eventually those to the SAA problems. The construction is used to study the consistency using results established in the literature on stochastic programming. The assumptions of our framework are verified on three nonlinear optimization problems under uncertainty.

Appendix A: Lack of Inf-Compactness

Besides the feasible set’s lack of compactness, the set of $\varepsilon$-optimal solutions to the SAA problem and the level sets of the SAA problem’s objective function may be noncompact for risk-neutral PDE-constrained optimization problems. We illustrate this observation on the semilinear PDE-constrained problem

$$\begin{aligned} \min _{u\in U_\text {ad}}\, (1/2){\mathbb {E}}\left[ \left\Vert S(u,\xi )\right\Vert _{L^2(D)}^2\right] +(\alpha /2)\left\Vert u\right\Vert _{L^2(D)}^2, \end{aligned}$$

(30)

where $\alpha > 0$, $U_\text {ad}= \{\, u \in L^2(D) :\left\Vert u\right\Vert _{L^2(D)} \le 2\}$, $D\subset {\mathbb {R}}^2$ is a bounded Lipschitz domain, and $\Xi$ is as in Sect. 3. For each $(u,\xi ) \in L^2(D) \times \Xi$, the state $S(u,\xi ) \in H_0^1(D)$ is the solution to the semilinear PDE: find $y \in H_0^1(D)$ with

$$\begin{aligned} ( \kappa (\xi )\nabla y, \nabla v )_{L^2(D)^2} +( y^3, v )_{L^2(D)} = ( u, v )_{L^2(D)} \;\; \text {for all} \;\; v \in H_0^1(D). \end{aligned}$$

(31)

We assume that $\kappa :\Xi \rightarrow L^\infty (D)$ is strongly measurable and that there exists $\kappa _{\min } > 0$ with $\kappa (\xi )\ge \kappa _{\min }$ for all $\xi \in \Xi$. The SAA problem of (30) is given by

$$\begin{aligned} \min _{u\in U_\text {ad}}\, \frac{1}{2N}\sum _{i=1}^N \left\Vert S(u,\xi ^i)\right\Vert _{L^2(D)}^2 +(\alpha /2)\left\Vert u\right\Vert _{L^2(D)}^2, \end{aligned}$$

(32)

where $\xi ^1, \xi ^2,\ldots$, are as in Sect. 3.

Let ${\hat{F}}_{N}$ be the objective function of (32) and let $C_D$ be Friedrichs ’ constant of the domain $D$. For each $(u,\xi ) \in L^2(D) \times \Xi$, we have the stability estimate (cf. [24, eqns. (2.11)])

$$\begin{aligned} \left\Vert S(u,\xi )\right\Vert _{H_0^1(D)} \le (C_D/\kappa _{\min })\left\Vert u\right\Vert _{L^2(D)}. \end{aligned}$$

(33)

The optimal value of the risk-neutral problem (30) and those of the corresponding SAA problems (32) are zero, as $S(0,\xi ) = 0$ for all $\xi \in \Xi$ and $0 \in U_\text {ad}$. We define $\varepsilon _{\max } = (C_D^2/\kappa _{\min })^2+\alpha$. Let $\varepsilon > 0$ satisfy $\varepsilon \le \varepsilon _{\max }$. We define $V_{\varepsilon } = \big \{\, u \in L^2(D) :\, \left\Vert u\right\Vert _{L^2(D)}^2 \le \tfrac{2\varepsilon }{(C_D^2/\kappa _{\min })^2+\alpha } \,\big \}$. It holds that $V_{\varepsilon } \subset U_\text {ad}$. For each $u \in V_\varepsilon$, the stability estimate (33) and Friedrichs ’ inequality yield

$$\begin{aligned}&\frac{1}{2N} \sum _{i=1}^N \left\Vert S(u,\xi ^i)\right\Vert _{L^2(D)}^2 + (\alpha /2)\left\Vert u\right\Vert _{L^2(D)}^2\\ {}&\quad \le (1/2)(C_D^2/\kappa _{\min })^2 \left\Vert u\right\Vert _{L^2(D)}^2 + (\alpha /2)\left\Vert u\right\Vert _{L^2(D)}^2 \\&\quad \le 0 + \varepsilon . \end{aligned}$$

Hence each $u \in V_\varepsilon$ is an $\varepsilon$-optimal solution to the SAA problem (32). The set $V_\varepsilon$ is a closed ball about zero with positive radius because $\varepsilon > 0$. Since $L^2(D)$ is infinite dimensional, this set is noncompact [49, Theorem 2.5-5]. Therefore, as long as ${\tilde{\varepsilon }} > 0$, the level sets of the SAA objective function, $\{ \, u \in U_\text {ad}:\, {\hat{F}}_{N}(u) \le {\tilde{\varepsilon }} \, \}$, are noncompact, as they contain the noncompact set $V_\varepsilon$ with $\varepsilon = \min \{{\tilde{\varepsilon }},\varepsilon _{\max }\}$. In this case, an inf-compactness condition (see [72, p. 166]) is violated.