Robust Stochastic Programming with Moment Uncertainty

Practically, one has limited information about the distribution $F$ driving the uncertain parameters that are involved in the decision-making process. In such situations, it might instead be safer to rely on estimates of the mean ${\mu }_0$ and covariance matrix ${\Sigma }_0$ of random vector—e.g. using empirical estimates. Howvever, we believe that in such problems, it is also rarely the case that one is entirely confident in these estimates. For this reason, we propose representing this uncertainty using two constraints parameterized by ${\gamma }_1\ge 0$ and ${\gamma }_2\ge 1$:

$$\begin{array}{rlr}& (\mathbb{E}[\xi ]-{\mu }_0{)}^T{\Sigma }_0^{-1}(\mathbb{E}[\xi ]-{\mu }_0)\le {\gamma }_1& (1a)\\ & \mathbb{E}[(\xi -{\mu }_0)(\xi -{\mu }_0{)}^T]\le {\gamma }_2{\Sigma }_0& (1b)\end{array}$$

Whereas constraint $(1a)$ assumes that the mean of $\xi$ lies in an ellipsoid of size ${\gamma }_1$ centered at the estimate ${\mu }_0$, constraint $(1b)$ forces $\mathbb{E}[(\xi -{\mu }_0)(\xi -{\mu }_0{)}^T]$, which we will refer to as the centered second-moment matrix of $\xi$, to lie in a positive semidefinite cone defined with a matrix inequality. In other words, it describes how likely $\xi$ is to be close to ${\mu }_0$ in terms of the correlations expressed in ${\Sigma }_0$. Finally, the parameters ${\gamma }_1$ and ${\gamma }_2$ provide natural means of quantifying one’s confidence in ${\mu }_0$ and ${\Sigma }_0$, respectively.

In what follows, we will study the DRSP under the distributional set
$${\mathcal{D}}_1(\mathcal{S},{\mu }_0,{\Sigma }_0,{\gamma }_1,{\gamma }_2)=\{\begin{array}{l}F\in \mathcal{M}\mid \begin{array}{rl}& \mathbb{P}(\xi \in \mathcal{S})=1\\ & (\mathbb{E}[\xi ]-{\mu }_0{)}^T{\Sigma }_0^{-1}(\mathbb{E}[\xi ]-{\mu }_0)\le {\gamma }_1\\ & \mathbb{E}[(\xi -{\mu }_0)(\xi -{\mu }_0{)}^T]\le {\gamma }_2{\Sigma }_0\end{array}\end{array}$$
where $\mathcal{M}$ is the set of all probability measures on the measurable space $({\mathbb{R}}^m,\mathcal{B})$, with $\mathcal{B}$ the Borel $\sigma$-algebra on ${\mathbb{R}}^m$ and $\mathcal{S}\subseteq {\mathbb{R}}^m$ is any closed convex set known ti contain the support of $F$.

Example: Distributionally robust optimization with piecewise-linear convex cost

Assume that one is interested in solving the following DRSP model for a general piecewise-linear convex cost function of $x$:
$$\underset{x\in \mathcal{X}}{\min }(\underset{F\in {\mathcal{D}}_1}{\max }{\mathbb{E}}_F[\underset{k}{\max }{\xi }_k^{T}x])$$
where each ${\xi }_k\in {\mathbb{R}}^n$ is a random vector. This model is quite applicable because convex cost functions can be approximated by piecewise-linear functions. By considering $\xi$ to be a random matrix whose $k$th column is the random vector ${\xi }_k$ and taking $h_k(x,\xi )={\xi }_k^{T}x$.

Example: Distributionally robust conditional Value-at-risk

Conditional value-at-risk (CVaR), also called mean excess loss, was recently introduced in the mathematical finance community as a risk measure for decision-making. It is closely related to the more common value-at-risk measure, which for a risk tolerance level of $\theta \in (0,1)$ evalutes the lowest amount $\tau$ such that with probability $1-\theta$, the loss does not exceed $\tau$. CVaR has gained a lot of interest in the community because of its attractive computational properties. For example, Rockafellar & Uryasev (2000) demonstrated that one can evaluate the $\theta$-CVaR of a cost function $c(x,\xi )$, where the random vector $\xi$ is distributed according to $F$, by solving a convex minimization problem:
$$\underset{\lambda \in \mathbb{R}}{\min }\lambda +\frac{1}{\theta }{\mathbb{E}}_F[(c(x,\xi )-\lambda {)}^{+}]$$
By the equivalence statement presented above, this problem is equivalent to the form
$$\underset{x\in \mathcal{X}}{\min }(\underset{F\in {\mathcal{D}}_1}{\max }(\underset{\lambda \in \mathbb{R}}{\min }\lambda +\frac{1}{\theta }{\mathbb{E}}_F[(c(x,\xi )-\lambda {)}^{+}]))$$