Two-Stage Distributionally Robust Conic Linear Programming over 1-Wasserstein Balls

TL;DR

This paper presents optimality conditions for the dual of the worst-case expectation problem, which characterizes worst-case uncertain parameters for its inner maximization problem, and highlights the potential advantage of a specific distance metric for out-of-sample performance.

Abstract

This paper studies two-stage distributionally robust conic linear programming under constraint uncertainty over type-1 Wasserstein balls. We present optimality conditions for the dual of the worst-case expectation problem, which characterizes worst-case uncertain parameters for its inner maximization problem. This condition offers an alternative proof, a counter-example, and an extension to previous works. Additionally, the condition highlights the potential advantage of a specific distance metric for out-of-sample performance, as exemplified in a numerical study on a facility location problem with demand uncertainty. A cutting-plane-based algorithm and a variety of algorithmic enhancements are proposed with a finite convergence proof under less stringent assumptions.

Figures (6)

• Figure 1: Three cases of inner supremum problem when $p=1$ for a box constrained $\Xi$; thick lines indicate the graph of the objective function $(\pi^T T(x))_j \xi_j - \lambda |\xi_j-\zeta^i_j|$ and red dots indicate the optimum for each case
• Figure 1: Estimated $f^T x + \mathbb E [Z(x,\xi)]$ using 2000 testing samples
• Figure 2: Potential support of the extremal distribution when $p=1$ and $p=2$ for a box-constrained $\Xi \subseteq \mathbb R^3$, indicated in blue
• Figure 2: Stochastic dominance of $\tilde{o}_{l_2}(\bar{\epsilon})$ to $\tilde{o}_{l_1}(\bar{\epsilon})$
• Figure 3: Support of the extremal distribution, Instance p21

Theorems & Definitions (24)

• Remark 1.1
• Lemma 2.2
• Proposition 3.1
• Theorem 3.2
• Proof 1
• Remark 3.3
• Corollary 3.4
• Proof 2
• Proposition 3.5
• Proof 3