Decision-Dependent Wasserstein Ambiguity Set

• The topic defines Wasserstein ambiguity sets whose centers and radii vary with decision variables, encoding endogenous uncertainty into the optimization model.
• It employs techniques like pushforward mappings, interpolation, and multimodal constructions to couple decision-making with distribution selection.
• Applications in pricing, wind-farm planning, and portfolio optimization demonstrate improved robustness, though challenges like nonconvexity and computational intensity remain.

A decision-dependent Wasserstein ambiguity set is a family of Wasserstein balls in which the ambiguity set itself varies with the decision variable, so that the admissible distributions are no longer exogenous but encode endogenous uncertainty. In the generic distributionally robust optimization formulation, one minimizes a cost of the form

$$\min_{x \in X}\left\{ f(x)+\sup_{P\in \mathcal{P}(x)} \mathbb{E}_{P}[h(x,\xi)]\right\},$$

with $\mathcal{P}(x)$ depending on $x$. In the Wasserstein case, the canonical form is

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$

where either the center $P_{\mathrm{ref}}(x)$, the radius $\epsilon(x)$, or both depend on the decision. This construction was formalized for decision-dependent distributionally robust optimization in early general form by Noyan, Rudolf, and Lejeune (Luo et al., 2018), and was subsequently specialized to pushforward-based, contextual, multimodal, interpolation-based, and application-specific models in later work (Fonseca et al., 2023, Fonseca et al., 2021, Zhu et al., 2024, Yu et al., 2024, Qu et al., 9 Aug 2025, Chen et al., 14 Aug 2025).

1. Formal definition and mathematical structure

In its most general form, the $p$-Wasserstein ambiguity set is

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$

with

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$

where $\Pi(P,Q)$ denotes the set of couplings and $\mathcal{P}(x)$0 is typically induced by a norm on $\mathcal{P}(x)$1 (Luo et al., 2018). The decision dependence may enter through a decision-dependent radius $\mathcal{P}(x)$2, through a decision-dependent reference measure $\mathcal{P}(x)$3, or through both.

Two mathematically distinct constructions recur in the literature. The first retains the original uncertainty space $\mathcal{P}(x)$4 and makes the ambiguity set depend on $\mathcal{P}(x)$5 directly, for example through $\mathcal{P}(x)$6 or $\mathcal{P}(x)$7. This is the organizing perspective in the original decision-dependent DRO framework and in interpolation-based and contextual constructions (Luo et al., 2018, Qu et al., 9 Aug 2025, Zhu et al., 2024). The second pushes the uncertainty through a decision-dependent map, typically $\mathcal{P}(x)$8 or $\mathcal{P}(x)$9, and places the Wasserstein ball on the scalar image distribution in $x$0. In that case, the center becomes the empirical pushforward $x$1 or $x$2, and the radius is scaled by a Lipschitz modulus such as $x$3 (Fonseca et al., 2023, Fonseca et al., 2021).

A central consequence of decision dependence is that the ambiguity set is endogenous: the decision changes not only the loss $x$4 but also the set of distributions against which robustness is enforced. In the terminology used in the foundational paper, these models arise in situations with endogenous uncertainty (Luo et al., 2018). Examples explicitly cited across the literature include pricing affecting demand, investment decisions changing failure probabilities or yields, sensor or infrastructure placement influencing error distributions, multimodal uncertainty with decision-dependent mode probabilities, and wind-farm planning in which capacity allocation changes the distribution of aggregated wind output through a smoothing effect (Luo et al., 2018, Yu et al., 2024, Chen et al., 14 Aug 2025).

2. Core distributionally robust formulations

The baseline decision-dependent DRO model is

$x$5

and in the Wasserstein specialization with fixed nominal center $x$6 and decision-dependent radius $x$7 this becomes

$x$8

(Luo et al., 2018). The same structure extends to two-stage formulations in which $x$9 is itself a recourse value function. In that setting,

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$0

and the outer problem becomes a two-stage decision-dependent DRO model (Luo et al., 2018).

Later papers specialized this template in several directions. A pushforward-based formulation places separate decision-dependent Wasserstein balls around empirical image distributions of objective and constraint functions, yielding

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$1

(Fonseca et al., 2021). A closely related unconstrained version robustifies the scalar random variable $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$2 and studies

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$3

(Fonseca et al., 2023).

In multimodal models, the ambiguity set is a mixture

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$4

where the mode probabilities are controlled by a decision-dependent $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$5-divergence set and the within-mode distributions by decision-dependent Wasserstein balls

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$6

(Yu et al., 2024). In contextual residual models, the nominal distribution is built from a learned mean $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$7 and empirical residuals, and the ambiguity set is

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$8

(Zhu et al., 2024). In interpolation-based dynamic pricing, the nominal distribution is

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$9

with Wasserstein ambiguity

$P_{\mathrm{ref}}(x)$0

(Qu et al., 9 Aug 2025).

These formulations share a common interpretation: the Wasserstein ball no longer encodes only finite-sample uncertainty around a fixed nominal law, but also the way the decision perturbs the nominal law or the robustness budget. This suggests that decision-dependent Wasserstein DRO is best viewed as a joint model of optimization and distribution selection rather than a mere robustification of a static stochastic program.

3. Principal forms of decision dependence

The literature exhibits three main patterns.

The first pattern is a decision-dependent radius with fixed center. In the original finite-support and continuous-support formulations, the reference distribution $P_{\mathrm{ref}}(x)$1 is fixed and the decision affects only the Wasserstein radius $P_{\mathrm{ref}}(x)$2 (Luo et al., 2018). The same idea appears in standard-quadratic optimization, where the ambiguity set is centered at the empirical distribution $P_{\mathrm{ref}}(x)$3 and uses a radius $P_{\mathrm{ref}}(x)$4, leading to

$P_{\mathrm{ref}}(x)$5

with the main results focusing on decision dependence only through $P_{\mathrm{ref}}(x)$6 (Bomze et al., 5 Mar 2026).

The second pattern is a decision-dependent center. In pushforward models, the center itself depends on $P_{\mathrm{ref}}(x)$7 because it is the empirical distribution of $P_{\mathrm{ref}}(x)$8 or $P_{\mathrm{ref}}(x)$9 (Fonseca et al., 2023, Fonseca et al., 2021). In multimodal DRO, each mode has a decision-dependent empirical distribution $\epsilon(x)$0, constructed by residuals-based sample generation (Yu et al., 2024). In dynamic pricing, the nominal distribution is obtained by interpolating empirical measures observed at different decisions, with weights satisfying $\epsilon(x)$1, $\epsilon(x)$2, and $\epsilon(x)$3 at sample decisions (Qu et al., 9 Aug 2025). In residuals-based contextual DRO, the center is the pushforward of the empirical residual distribution shifted by $\epsilon(x)$4 and optionally projected onto $\epsilon(x)$5 (Zhu et al., 2024).

The third pattern makes both center and effective radius decision dependent. Wind-planning under smoothing effects provides a direct example: the nominal empirical distribution is the empirical law of a decision-dependent aggregate such as

$\epsilon(x)$6

and the radius is calibrated as

$\epsilon(x)$7

or, for transmission aggregates,

$\epsilon(x)$8

(Chen et al., 14 Aug 2025).

A recurrent misconception is that “decision-dependent Wasserstein” necessarily means only a variable radius. The later literature shows that the center may be decision dependent as well, either by pushforward, residual shifting, multimodal conditioning, or interpolation (Fonseca et al., 2023, Yu et al., 2024, Zhu et al., 2024, Qu et al., 9 Aug 2025, Chen et al., 14 Aug 2025). Conversely, the early general framework explicitly notes that decision-dependent reference distributions are possible in principle, but its main Wasserstein development keeps $\epsilon(x)$9 fixed and lets decision dependence enter through $p$0 (Luo et al., 2018).

4. Reformulations, duality, and tractability

For finite support $p$1 with nominal distribution $p$2, the inner worst-case expectation over a $p$3-ball can be written as a linear program in $p$4: $p$5 subject to the transport-balance equations, the transport budget

$p$6

and probability constraints (Luo et al., 2018). Dualizing yields a finite-dimensional reformulation: $p$7 subject to

$p$8

with $p$9 and the remaining sign restrictions (Luo et al., 2018). The two-stage extension replaces $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$0 by $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$1 and adds recourse feasibility constraints $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$2 (Luo et al., 2018).

For continuous support, the same paper derives a conic-dual semi-infinite reformulation: $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$3 subject to

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$4

with $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$5 (Luo et al., 2018). This is a semi-infinite program because the constraints are indexed by $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$6.

In pushforward-based models, Kantorovich-type duality often collapses the worst-case expectation to explicit regularized empirical objectives. When $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$7 is an interval and $$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$8,

$$\mathcal{A}(x)=\{Q\in\mathcal{P}(\Xi):W_p(Q,P_{\mathrm{ref}}(x))\le \epsilon(x)\},$$9

and when $$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$0 is unbounded above,

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$1

(Fonseca et al., 2023). The expected-value-constrained variant extends this logic to both the objective and the constraint, producing finite-dimensional programs in which the robust lower bound for $$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$2 is the empirical mean minus a decision-dependent Lipschitz penalty (Fonseca et al., 2021).

A distinct but related tractability mechanism appears when the loss is linear in the uncertain parameter. In decision-dependent standard quadratic optimization, $$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$3 is linear in the uncertain matrix $$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$4, and under Euclidean/Frobenius geometry the worst-case expectation over a Wasserstein ball satisfies

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$5

so the DRO problem is exactly equivalent to

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$6

(Bomze et al., 5 Mar 2026). This shows that, in certain linear-in-uncertainty settings, the Wasserstein ambiguity set induces an explicit deterministic regularization.

In multimodal DRO, the within-mode worst-case expectation admits a Kantorovich-dual representation using conjugates and support functions, and the full problem becomes a monolithic reformulation once the mode-probability ambiguity is incorporated (Yu et al., 2024). In dynamic pricing, the discrete interpolated nominal measure produces a semi-infinite convex reformulation

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$7

subject to

$$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$8

(Qu et al., 9 Aug 2025).

5. Convexity, nonconvexity, and statistical calibration

A central theoretical divide concerns convexity. In the original decision-dependent Wasserstein framework, nonconvexity typically comes from bilinear coupling between a decision-dependent radius and a dual variable: $$W_p(P,Q)=\left(\inf_{\pi\in\Pi(P,Q)}\int_{\Xi\times\Xi} d(\xi,\zeta)^p\,d\pi(\xi,\zeta)\right)^{1/p},$$9 in finite support or $\Pi(P,Q)$0 in continuous support (Luo et al., 2018). This is one of the main distinctions from standard decision-independent Wasserstein DRO, whose dualized forms are typically convex in the outer variables when the underlying loss is convex (Luo et al., 2018).

By contrast, the pushforward-based constructions are often convex because the decision dependence is absorbed into Lipschitz moduli such as $\Pi(P,Q)$1 and empirical image distributions. In the unbounded-image case, the robust objective becomes an empirical mean plus the convex penalty $\Pi(P,Q)$2, with

$\Pi(P,Q)$3

and in portfolio models $\Pi(P,Q)$4 or $\Pi(P,Q)$5, leading to SOCP formulations (Fonseca et al., 2021). The unconstrained counterpart reaches the same conclusion for many losses $\Pi(P,Q)$6 whose Lipschitz constant is a norm or seminorm of $\Pi(P,Q)$7 (Fonseca et al., 2023).

Decision-dependent Wasserstein sets also admit explicit feasibility thresholds in expected-value-constrained models. Defining

$\Pi(P,Q)$8

and

$\Pi(P,Q)$9

feasibility in the unbounded-image case holds iff $\mathcal{P}(x)$00 and $\mathcal{P}(x)$01 (Fonseca et al., 2021).

Calibration of the ambiguity radius is treated differently across subliteratures. Interpolation-based DD-DRO uses

$\mathcal{P}(x)$02

where $\mathcal{P}(x)$03 is the covering radius of observed decisions in $\mathcal{P}(x)$04, and proves

$\mathcal{P}(x)$05

under Lipschitz assumptions on the interpolation and the true decision-to-distribution map (Qu et al., 9 Aug 2025). Residuals-based contextual DRO derives finite-sample radii

$\mathcal{P}(x)$06

from regression error bounds and Wasserstein concentration, yielding high-probability certificates of the form

$\mathcal{P}(x)$07

(Zhu et al., 2024). Wind-planning models exploit one-dimensional aggregation and log-concavity to calibrate a radius proportional to decision-dependent standard deviation, simplified in practice to

$\mathcal{P}(x)$08

(Chen et al., 14 Aug 2025).

These calibration results imply more than mere ambiguity-set construction. The interpolation-based framework proves non-asymptotic bounds

$\mathcal{P}(x)$09

with probability at least $\mathcal{P}(x)$10 (Qu et al., 9 Aug 2025). The wind-planning formulation states that the optimal objective value provides a probabilistic guarantee on out-of-sample performance, and explicitly derives a bound in terms of the regularized objective $\mathcal{P}(x)$11 (Chen et al., 14 Aug 2025). In residual contextual DRO, asymptotic optimality and rates of convergence are established under shrinking radii and regression conditions (Zhu et al., 2024).

6. Algorithms, applications, and limitations

Because decision dependence often destroys the clean convexity of ordinary Wasserstein DRO, the algorithmic picture is heterogeneous. The original framework recommends global nonlinear optimization techniques, including branch-and-bound or spatial branch-and-bound, for the resulting nonconvex programs, and proposes a cutting-surface algorithm for generic semi-infinite programs that terminates finitely with an $\mathcal{P}(x)$12-optimal solution under compactness of $\mathcal{P}(x)$13 and continuity of the constraint function (Luo et al., 2018). Interpolation-based DD-DRO likewise yields semi-infinite convex programs solved by cutting-surface methods, with finite convex reductions available in structured applications such as dynamic pricing (Qu et al., 9 Aug 2025).

Structured models often admit conic or mixed-integer conic formulations. Pushforward-based portfolio models become SOCPs in mean–variance and mean–CVaR forms (Fonseca et al., 2021). Multimodal DRO with affine decision-dependent mode probabilities and binary first-stage decisions can be reformulated as MILP for variation distance or MISOCP for $\mathcal{P}(x)$14-distance via McCormick envelopes (Yu et al., 2024). Wind-planning with one-dimensional aggregates and variance-based radii is reformulated as a mixed-integer second-order cone program; a constraint-generation framework and an L-shaped acceleration are reported to speed computation by “hundreds of times” in the numerical study (Chen et al., 14 Aug 2025).

Applications show why decision dependence matters. In facility location, multimodal decision-dependent DRO is reported to outperform single-modal and decision-independent baselines in in-sample and out-of-sample performance under the settings tested, and the paper proves that the multimodal ambiguity set is nested inside a grouped single-ball surrogate: $\mathcal{P}(x)$15 with corresponding weakly smaller worst-case cost (Yu et al., 2024). In dynamic pricing, decision-dependent nominal distributions are built from offline data under different historical prices, and the resulting DD-DRO produces pricing strategies with guaranteed expected revenue (Qu et al., 9 Aug 2025). In joint wind planning and operations, the ambiguity set is defined on decision-dependent one-dimensional aggregates so that diversified siting reduces the calibrated radius through the smoothing effect; the experiments report lower average risk-management costs and lower variance of aggregated wind output for DDRO relative to NDRO and empirical optimization in the tested instances (Chen et al., 14 Aug 2025). In portfolio optimization, decision-dependent image-space Wasserstein models yield convex formulations with strong empirical constraint satisfaction and explicit feasibility thresholds (Fonseca et al., 2021), while the related unconstrained alternative is reported to achieve similar out-of-sample behavior to standard DRO with simpler optimization in Mean–CVaR settings (Fonseca et al., 2023).

Several limitations are consistent across the literature. Decision dependence may induce nonconvexity through bilinear couplings, especially when the radius or the center depends directly on the optimization variable (Luo et al., 2018, Chen et al., 14 Aug 2025). Semi-infinite formulations remain computationally intensive when the support is continuous (Luo et al., 2018, Qu et al., 9 Aug 2025). Parameter estimation is itself nontrivial: constructing $\mathcal{P}(x)$16, $\mathcal{P}(x)$17, $\mathcal{P}(x)$18, or $\mathcal{P}(x)$19 requires statistical modeling assumptions whose reliability varies by application (Luo et al., 2018, Yu et al., 2024, Zhu et al., 2024, Qu et al., 9 Aug 2025). The contextual and interpolation-based approaches also inherit dimension dependence from Wasserstein concentration, and the interpolation framework states explicitly that the statistical term deteriorates with high-dimensional $\mathcal{P}(x)$20 (Qu et al., 9 Aug 2025).

A final misconception is that decision-dependent Wasserstein ambiguity sets are merely a technical variant of standard Wasserstein DRO. The collective evidence suggests a broader interpretation: they are a modeling device for endogenous uncertainty, and their main contribution is not only additional robustness but a change in what distributions are considered plausible after a decision is chosen (Luo et al., 2018, Zhu et al., 2024, Chen et al., 14 Aug 2025).