Wasserstein Ambiguity Ball in Robust Optimization

• Wasserstein ambiguity ball is an uncertainty set defined via the Wasserstein distance, measuring the gap between probability distributions for robust optimization.
• It enables data-driven, finite-sample performance guarantees by tuning the ball’s radius to mitigate sampling, estimation, and adversarial errors.
• Conic and linear reformulations of DRO models using the Wasserstein ball facilitate practical applications in robust regression, inventory management, and decision-making.

A Wasserstein ambiguity ball is an uncertainty set in probability distribution space defined via the Wasserstein metric, which quantifies the “distance” between probability measures. Its central role is in modeling distributional uncertainty in stochastic optimization, statistical learning, and robust decision-making, where the objective or constraints must hold robustly over all distributions within a specified ball around a nominal law. The flexibility to tune the center and radius enables a rigorous, data-driven, and often finite-sample–certified approach for hedging against sampling error, estimation error, and adversarial perturbations.

1. Mathematical Definition and Model-Theoretic Role

For a measurable metric space $(\Xi, d)$, and $r \geq 1$, the $r$-Wasserstein distance between $P_1, P_2 \in \mathcal{M}_r(\Xi)$ is defined as

$$W_r(P_1, P_2) = \inf_{Q \in \Gamma(P_1, P_2)} \left( \int_{\Xi \times \Xi} d(\xi_1, \xi_2)^r Q(d\xi_1, d\xi_2) \right)^{1/r}$$

where $\Gamma(P_1, P_2)$ is the set of all couplings with marginals $P_1, P_2$. The Wasserstein ambiguity ball of radius $\epsilon$ centered at a reference distribution $\widehat{P}_I$ is

$$B_r(\epsilon; \widehat{P}_I) = \left\{ P \in \mathcal{M}_r(\Xi): W_r(P, \widehat{P}_I) \leq \epsilon \right\}$$

This set is the closed ball under $W_r$ around the empirical (or otherwise nominal) measure. It serves as an ambiguity set for distributionally robust optimization (DRO) or learning procedures, which then seek to minimize (or constrain) the worst-case objective evaluated over all $P$ in the ball: $$\min_{x \in \mathcal{X}} \sup_{P \in B_r(\epsilon; \widehat{P}_I)} \mathbb{E}_P \left[ f(x, \xi) \right]$$ The radius $\epsilon$ quantifies the level of ambiguity aversion, with $\epsilon=0$ collapsing the model to empirical/stochastic optimization, while $\epsilon \to \infty$ recovers purely robust (worst-case) formulations (Hanasusanto et al., 2016).

2. Finite-Dimensional and Conic Reformulations

The infinite-dimensional worst-case expectation over a Wasserstein ambiguity ball can often be recast as a tractable finite-dimensional program. For two-stage distributionally robust linear programs (DRO-LPs) with recourse, the paper (Hanasusanto et al., 2016) establishes that the key recourse function

$$\mathcal{Z}(x) = \sup_{P \in B_r(\epsilon; \widehat{P}_I)} \mathbb{E}_P [Z(x, \xi)]$$

is equivalent to a generalized moment problem, which, via strong duality, can be exactly reformulated for $r=2$ as a conic (specifically, copositive) program under complete recourse. The core dual representation involves auxiliary transportation variables and Lagrangian relaxation of the moment (distance) constraints, leading to conic problems of the form: $$\min \ \epsilon^2 \lambda + \frac{1}{I} \sum_{i \in [I]} s_i + \cdots \quad \text{s.t. conic constraints} \quad [\cdot] \succeq_{\mathcal{C}} 0$$ with $\mathcal{C}$ the copositive cone. The dual problem is a completely positive program. Strong duality holds under complete recourse.

For $r=1$ and under no support constraints with only constraint right-hand side uncertainty, the corresponding DRO problem collapses to a tractable linear program (Hanasusanto et al., 2016).

3. Copositive Programs and Hierarchies of Relaxations

When reducing DRO-LPs over $2$-Wasserstein balls to copositive programs, exact equivalence is achieved under complete recourse, while in the case of expensive or incomplete recourse, arbitrarily close approximations can be made by a sequence of copositive relaxations (Hanasusanto et al., 2016). As optimization over the full copositive cone is NP-hard, the authors advocate tractable semidefinite inner approximations, e.g.,

$$\mathcal{C}^0 = \{ M \in S^K: M = P + N, \ P \succeq 0, \ N \geq 0 \}$$

This approximation is exact for $K \leq 4$, and safe for larger instances. More refined hierarchies (such as those by Parrilo, Bomze, etc.) can be used for tighter control of relaxation quality.

4. Relationship to Classical Robust and Stochastic Optimization

A Wasserstein ambiguity ball enables a unified framework interpolating between classical stochastic and robust optimization. When $\epsilon = 0$, only the nominal (empirical) measure is considered, recovering standard stochastic programming. For sufficiently large $\epsilon$ that the ball contains every Dirac measure on the support, the problem coincides with the pure robust formulation: $\min_{x} c^T x + \max_{\xi \in \Xi} Z(x, \xi)$ The conic reformulations developed for the Wasserstein ball extend seamlessly to the robust case by parameter selection in the appropriate limit, yielding polynomially sized, strong robust optimization formulations (Hanasusanto et al., 2016).

5. Special Case: 1-Wasserstein Ball and Linear Program Reformulations

For $r=1$, when the ambiguity set is a $1$-Wasserstein ball and the uncertainty is in the constraint right-hand side, the two-stage DRO problem admits a remarkable simplification: it can be reformulated as a tractable linear program provided that there are no support constraints (i.e., $\Xi = \mathbb{R}^K$) and the relevant norm is $L^1$ or a suitable weighted $L^1$ (Hanasusanto et al., 2016). Notably, the worst-case expectation then reduces to a supremum characterized by a linear programming duality, enabling exact, efficient solution strategies in settings such as robust regression or multi-item newsvendor problems.

6. Practical Implications, Applications, and Empirical Performance

Exact and approximate conic/LP reformulations make Wasserstein ambiguity-based DRO applicable in various domains:

• Robust Regression and LASSO: In least absolute deviations (LAD) regression, the distributionally robust formulation under a Wasserstein ambiguity set recovers the empirical loss plus an $L^1$-regularization term, thus providing a direct link to LASSO-style regularization as a byproduct of ambiguity modeling.
• Inventory/Newsvendor Models: For multi-item newsvendor problems, conic or linear reformulations are directly applicable for both $2$- and $1$-Wasserstein balls.
• Empirical Performance: Numerical experiments presented compare Wasserstein ball-based policies with those from Chebyshev sets and standard Sample Average Approximation (SAA), demonstrating that Wasserstein DRO solutions achieve superior out-of-sample performance and reduced optimality gaps, especially with limited data (Hanasusanto et al., 2016).

Model Class	Wasserstein Ball $r$	Reformulation Type	Tractability Condition
2-stage DRO-LP	$r=2$	Copositive/semidefinite program	Complete/sufficient recourse
2-stage DRO-LP	$r=1$	Linear program	No support constraints; $Q=0$
Robust regression	$r=1$/$r=2$	LP (LAD), SDP (LASSO)	Standard settings

7. Theoretical and Computational Impact

The deployment of Wasserstein ambiguity balls enables:

• Rigorous distributional uncertainty modeling with finite-sample performance guarantees, as the ball radius can be tied to confidence levels via concentration inequalities.
• Polynomial-size reformulations via conic programming, overcoming the dimensionality bottleneck typically associated with robust or DRO models.
• Stronger solution quality relative to classical robust or SAA approaches, especially in data-driven scenarios with limited or uncertain data.
• Structural unification of stochastic and robust programming through a single tunable ambiguity parameter, $\epsilon$, which serves as a regularization knob.

These contributions drive both theoretical advances in robust optimization and practical advances in large-scale applications, from machine learning to supply chain and financial decision making.