Statistical robustness in utility preference robust optimization models

Abstract

Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in the PRO models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution. In data-driven optimization, this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the PRO models based on perceived data are reliable. In this paper, we investigate the issue which is also known as qualitative robustness in the literature of statistics (Huber in Robust statistics, 3rd edn, Wiley, New York, 1981) and risk management (Krätschmer et al. in Finance Stoch 18:271–295, 2014). By utilizing the framework proposed by Krätschmer et al. (2014), we derive moderate sufficient conditions under which the optimal value and optimal solution of the PRO models are robust against perturbation of the exogenous uncertainty data, and examine how the tail behaviour of utility functions affects the robustness. Moreover, under some additional conditions on the Lipschitz continuity of the underlying functions with respect to random data, we establish quantitative robustness of the statistical estimators under the Kantorovich metric. Finally, we investigate uniform consistency of the optimal value and optimal solution of the PRO models. The results cover utility selection problems and stochastic optimization problems as special cases.

Appendix

Theorem 5

(Berge’s maximum theorem, Page 116 [3]) Let $\phi$ be a continuous numerical function in a topological space $X \times Y$ and $\varGamma$ be a continuous mapping of X into Y such that for each x, $\varGamma (x) \ne \emptyset$, then the numerical function M defined by $M(x):=\max \{\phi (x,y): y\in \varGamma (x)\}$ is continuous in X and the mapping $\varPhi$ defined by $\varPhi (x):=\{y| y\in \varGamma (x),\phi (y)=M(x)\}$ is a upper semicontinuous mapping of X into Y.

Proof of Theorem 2

Part (i). It follows from Theorem 1 (ii) that ${\vartheta }$ is continuous at P w.r.t. $\phi$-weak topology, that is, $|{\vartheta }(P')-{\vartheta }(P)| \rightarrow 0$ as $\mathsf {dl}_{\phi }(P',P) \rightarrow 0$. Together with the UGC property of $({{\mathcal {M}}}_{k,\kappa }^{\phi ^p},\mathsf {dl}_{\phi })$ (see [32, Corallary 3.5]), we can obtain the conclusion by applying [32, Theorem 2.4]. Here we include a sketch of proof for completeness. By triangle inequality

$$\begin{aligned} \mathsf {dl}_{\text{ Prok }}\left( P^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}, Q^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}\right)\le & {} \mathsf {dl}_{\text{ Prok }}\left( P^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}, \delta _{{\vartheta }(P)}\right) \nonumber \\&+\mathsf {dl}_{\text{ Prok }}\left( \delta _{{\vartheta }(P)}, Q^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}\right) , \end{aligned}$$

it suffice to show that for any small number $\epsilon >0$ there exist positive numbers $\delta >0$ and $N_0\in {\mathbb {N}}$ such that for all $N\ge N_0$

$$\begin{aligned} \mathsf {dl}_{\text{ Prok }}\left( P^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}, \delta _{{\vartheta }(P)}\right)\le & {} \frac{\epsilon }{2},\\ \mathsf {dl}_{\text{ Prok }}\left( \delta _{{\vartheta }(P)}, Q^{\otimes N}\circ {\hat{{\vartheta }}}_N^{-1}\right)\le & {} \frac{\epsilon }{2} \end{aligned}$$

when $\mathsf {dl}_{\phi }(P,Q) \le \delta$. By the Strassen’s theorem [27, Theorem 2.13], the above two inequalities are guaranteed respectively by

$$\begin{aligned} P^{\otimes N}\left[ \xi \in (\mathrm{I\!R}^k)^{\otimes N}: \left| {\vartheta }(P)-{\vartheta }(P_N) \right| \le \frac{\epsilon }{2} \right] \ge 1-\frac{\epsilon }{2}, \end{aligned}$$

(86)

and

$$\begin{aligned} Q^{\otimes N}\left[ {\tilde{\xi }}\in (\mathrm{I\!R}^k)^{\otimes N}: \left| {\vartheta }(P)-{\vartheta }(Q_N) \right| \le \frac{\epsilon }{2} \right] \ge 1-\frac{\epsilon }{2}, \end{aligned}$$

(87)

where $P_N$ and $Q_N$ are defined in (9). Since ${\vartheta }$ is continuous at P by Theorem 1, for any $\epsilon >0$, there exists a constant $\delta >0$ such that

$$\begin{aligned} \mathsf {dl}_{\phi }(P,P') \le 2\delta \Longrightarrow |{\vartheta }(P)-{\vartheta }(P')| \le \epsilon /2. \end{aligned}$$

(88)

By plugging $P_N$ into the position of $P'$, we obtain

$$\begin{aligned} P^{\otimes N}\left[ \xi \in (\mathrm{I\!R}^k)^{\otimes N}: \mathsf {dl}_{\phi } (P,P_N)\le 2\delta \right] \le P^{\otimes N}\left[ \xi \in (\mathrm{I\!R}^k)^{\otimes N}: \left| {\vartheta }(P)-{\vartheta }(P_N)\right| \le \frac{\epsilon }{2} \right] . \end{aligned}$$

Moreover, by the UGC property of $({{\mathcal {M}}}_{k,\kappa }^{\phi ^p},\mathsf {dl}_{\phi })$, there exists $N_0\in {\mathbb {N}}$ such that

$$\begin{aligned} P^{\otimes N}\left[ \xi \in (\mathrm{I\!R}^k)^{\otimes N}: \mathsf {dl}_{\phi } (P,P_N)\le \delta \right] \ge 1-\epsilon /2 \end{aligned}$$

for $N \ge N_0$, a combination of the two inequalities gives rise to (86). Next, we prove (87). For any fixed Q satisfying $\mathsf {dl}_{\phi }(P,Q) \le \delta$, we have $\mathsf {dl}_{\phi }(P,Q_N) \le \mathsf {dl}_{\phi }(P,Q) + \mathsf {dl}_{\phi }(Q,Q_N)$, By the UGC property

$$\begin{aligned} 1-\frac{\epsilon }{2}\le & {} Q^{\otimes N}\left[ {\tilde{\xi }}\in (\mathrm{I\!R}^k)^{\otimes N}: \mathsf {dl}_{\phi } (Q,Q_N)\le \delta \right] \\\le & {} Q^{\otimes N}\left[ {\tilde{\xi }}\in (\mathrm{I\!R}^k)^{\otimes N}: \mathsf {dl}_{\phi } (P,Q_N)\le \mathsf {dl}_{\phi }(P,Q) + \delta \right] \\\le & {} Q^{\otimes N}\left[ {\tilde{\xi }}\in (\mathrm{I\!R}^k)^{\otimes N}: \mathsf {dl}_{\phi } (P,Q_N)\le 2\delta \right] \\\le & {} Q^{\otimes N}\left[ {\tilde{\xi }}\in (\mathrm{I\!R}^k)^{\otimes N}: \left| {\vartheta }(P)-{\vartheta }(Q_N)\right| \le \frac{\epsilon }{2}\right] , \end{aligned}$$

which implies (87).

Part (ii). If for each $Q \in {\mathcal {M}}$, S(Q) and $S(Q_N)$ are singletons, it follows from Theorem 1 (iii) that $S(\cdot )$ is continuous at P w.r.t. $\phi$-weak topology. The rest holds based on similar analysis in Part (i).$\square$