Distributionally robust optimization through the lens of submodularity

TL;DR

The paper advances distributionally robust optimization by introducing a submodular ambiguity set that naturally captures marginal and dependence information for both discrete and continuous uncertainty. It shows that, for discrete uncertainty, the inner worst-case expectation can be solved in polynomial time via duality and submodular minimization, and provides a compact polynomial-size linear program for structured univariate and bivariate information. For continuous uncertainty, it achieves additive-error solvability in pseudo-polynomial time through discretization and a structured semidefinite program when moment information is specified, offering a tractable relaxation where exact covariance specification would be NP-hard. Numerical experiments on project networks and multi-newsvendor problems demonstrate the modeling flexibility and computational practicality of the approach, underscoring the submodular ambiguity set as the natural discrete counterpart to convex ambiguity sets and a complement to them for continuous uncertainty.

Abstract

Distributionally robust optimization is used to tackle decision making problems under uncertainty where the distribution of the uncertain data is ambiguous. Many ambiguity sets have been proposed for continuous uncertainty that build on convexity and for which the resulting formulations scale polynomially in the number of random variables. However fewer ambiguity sets have been proposed for discrete uncertainty where the exact formulations scale polynomially in the number of random variables. Towards this, we define a submodular ambiguity set and showcase its expressive power in modeling both discrete and continuous uncertainty. With discrete uncertainty, we show that a class of distributionally robust optimization problems is solvable in polynomial time by viewing it through the lens of submodularity. With continuous uncertainty, we show that it is solvable approximately up to an additive error in pseudo-polynomial time. We then focus on a specific class of submodular ambiguity sets where univariate marginal information and bivariate dependence information on the random vector is specified and provide an exact reformulation as a polynomial sized linear program when the uncertainty is discrete and as a polynomial sized semidefinite program when the uncertainty is continuous. We provide numerical evidence of the modeling flexibility and expressive power of the submodular ambiguity set and demonstrate its applicability in two examples: project networks and multi-newsvendor problems. The paper highlights that the submodular ambiguity set is the natural discrete counterpart of the convex ambiguity set and supplements it for continuous uncertainty, both in modeling and computation.

Figures (7)

• Figure 1: Percentage relative difference in objective values given by $\frac{(\rho_{\text{rel}}^*- \rho_{\text{eq}}^*)}{\rho_{\text{eq}}^*}*100\%$ for varying values of $\alpha,\beta$.
• Figure 2: Comonotonic construction for the conditional distributions in step (ii) for $N = 2$ and $K = 2$. Here the solid line indicates $\tilde{\xi}_1 \sim F_{1|k}^{-1}(U)$ and the dashed line indicates $\tilde{\xi}_2 \sim F_{2|k}^{-1}(U)$ for $k = 1$ (left figure) and $k = 2$ (right figure) where $F_{i|k}$ is the conditional marginal distribution of $\tilde{\xi}_i$ for index $k$ being optimal.
• Figure 3: Subfigures (a) and (b) display the support of the conditional bivariate distributions for $k = 1$ and $k = 2$ while (c) shows that overall support of the extremal bivariate distribution using the weighted probabilities $\lambda_1^*$ and $\lambda_2^*$. While the conditional bivariate distributions in (a) and (b) are comonotonic, the final bivariate distribution in (c) is not comonotonic but positively dependent here.
• Figure 4: PERT network with seven activities and five paths from start node $1$ to end $4$.
• Figure 5: Fairness-utility tradeoff for all $\alpha \in [0,1]$ where $\mathbb{P}^*_{\text{mm}}(\hbox{\boldmath $q$}_{\text{mm}}^*)$ and $\mathbb{P}^*_{\text{cm}}(\hbox{\boldmath $q$}_{\text{cm}}^*)$ correspond to results with the ambiguity set being correctly specified and $\mathbb{P}^*_{\text{cm}}(\hbox{\boldmath $q$}_{\text{mm}}^*)$ and $\mathbb{P}^*_{\text{mm}}(\hbox{\boldmath $q$}_{\text{cm}}^*)$ correspond to results with the ambiguity set being mis-specified.

Theorems & Definitions (18)

• Example 1: Bernoulli with lower bounds on covariances
• Example 2: Lower bounds on bivariate tail probabilities or cross moments
• Example 3: Upper bounds on pairwise Wasserstein distances
• Example 4: Upper bounds on expected norm of a nonnegative random vector
• Example 5: Upper bounds on expected value of concave functions of sum of random variables
• Example 6: Lower bounds on orthant probabilities
• Theorem 4.1
• proof
• Example 7: Extremal distribution for maximum of two Bernoulli random variables
• Example 8: Inequality constraints in the ambiguity set are not always tight for submodular objective