Shared Distributionally Robust Chance Constraints

• Shared DRCCs are chance constraints that use a common ambiguity set to enforce joint probabilistic guarantees across various system components.
• They leverage methods such as moment-based, Wasserstein, and Bayesian formulations to share risk and uncertainty among multiple constraints, time steps, or agents.
• Tractable optimization is enabled through convex reformulations and surrogate approaches like worst-case CVaR and polyhedral approximations despite inherent nonconvexity.

Shared distributionally robust chance constraints (DRCCs) are chance constraints whose probabilistic guarantee is enforced against a family of distributions rather than a single known law, and whose robustness is shared across multiple constraints, time steps, outputs, agents, or objective terms through a common ambiguity set or a common probability budget. In the recent literature, this shared structure appears as moment-based safety constraints that must hold for all distributions sharing prescribed moments, horizon-wide output constraints under one Wasserstein ball, joint security constraints across all lines and contingencies, and common feasibility constraints in generalized Nash games and Markov decision processes (Ryu et al., 2024, Coulson et al., 2020, Maghami et al., 2022, Wen et al., 17 Sep 2025).

1. Definition and structural variants

A classical chance constraint has the form

$\mathbb{P}\big(g(x,\xi)\le 0\big)\ge 1-\epsilon,$

where $x$ is the decision variable, $\xi$ is a random vector, and $\epsilon$ is the admissible violation probability. A distributionally robust chance constraint replaces the single distribution by an ambiguity set $\mathcal{P}$ and requires

$$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$

The guarantee is therefore stated against the worst distribution in $\mathcal{P}$, not merely against a nominal law (Cherukuri et al., 2020).

The literature uses the adjective “shared” in several closely related senses. One is the joint-event sense,

$$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$

where a single probability budget is shared by all inequalities. A second is the shared-budget sense,

$$\sum_{i=1}^m \alpha_i\le \epsilon,\qquad \inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)>0\big)\le \alpha_i,$$

where the global risk level is allocated across several DRCCs. A third is the shared-ambiguity sense, where one ambiguity set is used simultaneously by multiple constraints and often by the objective as well (Cherukuri et al., 2020).

This shared structure is explicit in several application areas. In distributionally robust DeePC, a single Wasserstein ball is shared across the worst-case expected cost and a single horizon-wide DRCC on the stacked output trajectory. In security-constrained dispatch, one Wasserstein ball is shared across all reserve and line-flow constraints, which are enforced jointly. In Bayesian DRCCs, a single credible set in parameter space is shared across multiple constraints together with a risk-budget allocation $\sum_j \epsilon_j=\bar\epsilon$. In Wasserstein GNEPs, the shared DRCC couples all agents through one common feasibility condition (Coulson et al., 2020, Maghami et al., 2022, Chen et al., 2023, Wen et al., 17 Sep 2025).

2. Ambiguity sets and what is being shared

The object being shared is usually the ambiguity set. Different papers instantiate that set in different ways.

Ambiguity model	Representative definition	Shared role
Moment-based	$x$0	All admissible distributions share mean and covariance
Wasserstein	$x$1	One data-driven ball shared across constraints and often objective
Bayesian credible	$x$2	One posterior credible region shared across all parametric laws
Discrepancy / PRL	$x$3	One ambiguity radius determines a perturbed nominal risk level

A moment-based shared ambiguity set is used in orbital debris collision avoidance. At each prediction step $x$4, the set $x$5 contains all debris-position distributions that share the same estimated mean $x$6 and covariance $x$7. The resulting DRCC requires the collision probability bound to hold for every distribution consistent with those shared moments, which is exactly the moment-sharing interpretation of shared DRCCs (Ryu et al., 2024).

A Wasserstein ambiguity set is the dominant data-driven construction in the recent control and optimization literature. In distributionally robust DeePC, the ambiguity set is a Wasserstein ball around the empirical distribution of a random vector built from the Page or Hankel output matrices, and that single ball is shared across the worst-case expected cost and the robust CVaR constraint on the full output trajectory. In security-constrained dispatch, the same Wasserstein ball is shared across all reserve and line constraints inside a single joint chance constraint. In the general consistency theory for DRCCPs, the data-driven set is

$x$8

and the DRCC is imposed against every $x$9 (Coulson et al., 2020, Maghami et al., 2022, Cherukuri et al., 2020).

Bayesian DRCCs use a parametric ambiguity set in parameter space rather than a nonparametric ball in distribution space. The ambiguity set is a Bayesian credible region $\xi$0, which induces a family of laws $\xi$1. For joint constraints, this same credible region is shared across all component constraints, and the total risk $\xi$2 is split into $\xi$3 with $\xi$4 (Chen et al., 2023).

A different line of work uses ambiguity metrics through perturbed risk levels. For an ambiguity set

$\xi$5

the perturbed risk level $\xi$6 is defined so that a nominal chance constraint at level $\xi$7 implies the robust chance constraint at level $\xi$8. For the relative variation distance,

$\xi$9

so the robust risk budget is shared with ambiguity by a simple rescaling (Heinlein et al., 2024).

3. Reformulations and convex approximations

The central technical challenge is that exact DRCCs are usually nonconvex and difficult to solve directly. The literature therefore develops deterministic reformulations, conservative surrogates, and inner approximations.

A standard route passes through VaR and CVaR. In the orbital debris formulation, the chance constraint

$\epsilon$0

is equivalent to

$\epsilon$1

while

$\epsilon$2

is a conservative sufficient condition because $\epsilon$3. Under the moment-based ambiguity set and an ellipsoidal safety cost

$\epsilon$4

the worst-case CVaR admits the closed form

$\epsilon$5

so the DRCC is enforced by the deterministic trace inequality

$\epsilon$6

This is a particularly clear instance of a shared DRCC reducing to a low-dimensional convex condition involving only shared covariance information (Ryu et al., 2024).

In distributionally robust DeePC, the DRCC is implemented through a worst-case CVaR constraint on the full stacked future output,

$\epsilon$7

Under convexity and Lipschitz assumptions, the semi-infinite distributional problem is upper bounded by a finite convex program with a sample-average term and a norm regularization term $\epsilon$8. The shared ambiguity set thus appears in the tractable model only through empirical averages and norm penalties (Coulson et al., 2020).

For Wasserstein DR joint chance constraints in power dispatch, exact reformulations are computationally heavy, so a two-step method is used. First, a polyhedral uncertainty set $\epsilon$9 is constructed so that

$\mathcal{P}$0

Second, a robust linear program enforces all operational constraints for every $\mathcal{P}$1. Because $\mathcal{P}$2 carries enough mass under every distribution in the shared Wasserstein ball, any robustly feasible solution is feasible for the original shared DRCC (Maghami et al., 2022).

The general Wasserstein DRCCP literature provides two complementary viewpoints. One is an exact distance-to-violation reformulation based on

$\mathcal{P}$3

leading to sample-based deterministic representations of worst-case violation probability. The other is a convex inner approximation based on CVaR, which yields tractable convex programs when $\mathcal{P}$4 is convex and, in affine cases, explicit conic or mixed-integer reformulations (Hota et al., 2018).

Recent work on better convex approximations shows that the lower-level ALSO-X hinge-loss problem is a special case of the lower-level CVaR approximation, while the upper-level CVaR approximation is more restricted than the upper level in ALSO-X. This motivates ALSO-X#, which uses a lower-level CVaR approximation together with the less restricted ALSO-X upper-level chance constraint. Under Wasserstein ambiguity, ALSO-X# is always better than the CVaR approximation and, under additional uniqueness conditions, can be better than ALSO-X (Jiang et al., 2023).

In stochastic MPC, DRCCs are unified with probabilistic reachable sets. A distributionally robust probabilistic reachable set $\mathcal{P}$5 satisfies

$\mathcal{P}$6

and deterministic constraint tightening via Pontryagin difference then enforces the original DRCCs on states and inputs. This is another shared construction: one DR-PRS is reused at every time step (Mark et al., 2020).

4. Shared DRCCs across application domains

In orbital collision avoidance, the shared structure is temporal and moment-based. Safety must hold at every prediction step for all debris-position distributions sharing the propagated mean and covariance. The model predictive controller then trades fuel against a collision-risk bound, and smaller $\mathcal{P}$7 produces more conservative maneuvers, larger minimum satellite–debris separation, and higher total $\mathcal{P}$8 (Ryu et al., 2024).

In data-enabled predictive control of unknown stochastic LTI systems, the shared structure is horizon-wide and data-driven. The function $\mathcal{P}$9 acts on the entire stacked output trajectory, so the DRCC is a single joint constraint over all outputs and all future times, while the same Wasserstein ball is shared by both the output constraint and the worst-case expected objective (Coulson et al., 2020).

In security-constrained dispatch, the shared structure is system-wide. One joint DRCC covers reserve activation limits and line-flow limits for all lines and all contingencies, with

$$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$0

inequalities inside the chance constraint. This shared form is natural in power-system reliability because a single probability guarantee is sought for the entire security set rather than for separate constraints in isolation (Maghami et al., 2022).

In stochastic MPC, the shared structure is time-shared and set-based. A single stationary DR-PRS can tighten nominal constraints at all prediction steps, so the same confidence region is reused throughout the closed loop. The paper explicitly interprets this as DRCCs “shared over time” and also “shared” across halfspaces through allocation of individual violation probabilities whose sum does not exceed the global risk budget (Mark et al., 2020).

In Bayesian DRCCs, the shared structure is both statistical and combinatorial. A single posterior credible set defines all admissible distributions, and a joint BDRCC

$$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$1

is approximated by per-constraint BDRCCs with $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$2. The corresponding robust counterpart uses one uncertainty-set construction per $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$3 and proves that any feasible solution satisfies the true joint DRCC with probability $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$4 (Chen et al., 2023).

In Markov decision processes and generalized Nash games, the shared structure couples decisions. The distributionally robust joint chance-constrained MDP uses a product of individual robustness levels $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$5 with $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$6, which shares the probability budget across constraints. In GNEPs, the shared DRCC appears directly as one feasibility condition involving all agents’ strategies and one Wasserstein ambiguity set, which transforms the game into a generalized Nash equilibrium problem (Xia et al., 2023, Wen et al., 17 Sep 2025).

Black-box optimization offers a different shared viewpoint. Distributionally robust Bayesian optimization evaluates both the objective

$$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$7

and the chance quantity

$$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$8

over the same ambiguity set $$\inf_{Q\in\mathcal{P}} Q\big(g(x,\xi)\le 0\big)\ge 1-\epsilon.$$9, so objective and feasibility share the same distributional uncertainty model (Inatsu et al., 2022).

5. Guarantees, consistency, and statistical interpretation

The basic guarantee of every DRCC is conditional on ambiguity-set validity: if the true law belongs to the ambiguity set, then enforcing the DRCC implies the original chance constraint. This implication is explicit in the moment-based orbital setting and underlies the general Wasserstein DRCCP formulation as well (Ryu et al., 2024, Cherukuri et al., 2020).

For Wasserstein ambiguity sets, asymptotic consistency has been established for both DRCCPs and DR CVaR-constrained problems. If the samples are i.i.d., the ambiguity radii $\mathcal{P}$0 satisfy

$\mathcal{P}$1

and the technical regularity assumptions hold, then the optimal values of the distributionally robust problems converge almost surely to those of the corresponding true chance- or risk-constrained problems, and every accumulation point of the robust optimizers is an optimizer of the true problem (Cherukuri et al., 2020).

Finite-sample out-of-sample guarantees are central in data-driven control. In distributionally robust DeePC, a measure-concentration result provides a radius $\mathcal{P}$2 such that

$\mathcal{P}$3

With that radius, the optimizer of the tractable reformulation satisfies

$\mathcal{P}$4

and the robust objective upper-bounds the true expected cost with the same confidence (Coulson et al., 2020).

Bayesian DRCCs provide a different finite-sample interpretation. If the true parameter $\mathcal{P}$5 lies in the credible region $\mathcal{P}$6 with probability $\mathcal{P}$7, then any solution satisfying the BDRCC over that region satisfies the corresponding true chance constraint with probability at least $\mathcal{P}$8. For joint constraints, Theorem 3.3 shows that the risk-budgeted robust counterpart implies the true joint DRCC with probability $\mathcal{P}$9 (Chen et al., 2023).

Randomized methods under ambiguity admit one-level and two-level guarantees through perturbed risk levels. If a randomized scheme has nominal violation bound $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$0, then the robust version satisfies

$$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$1

For the scenario approach, this yields explicit distributionally robust guarantees and expected-violation bounds, especially simple under relative variation distance (Heinlein et al., 2024).

In stochastic MPC, DR-PRSs provide a time-uniform guarantee: if $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$2, then the DR-PRS covers the true PRS with high confidence and the tightened closed-loop MPC inherits the intended DRCC properties for all future times (Mark et al., 2020).

6. Conservatism, computation, and unresolved issues

A recurring misconception is that shared DRCCs are merely collections of independent per-constraint DRCCs. Several papers state the opposite: a shared DRCC may be a single joint chance constraint on a stacked trajectory, a single ambiguity set shared across objective and constraints, or a system-wide security event shared across all lines and contingencies. This suggests that “shared” refers to coupling in the probabilistic model, not merely to repeated notation (Coulson et al., 2020, Maghami et al., 2022).

A second misconception is that CVaR reformulations are exact representations of shared DRCCs. In the surveyed formulations, CVaR is generally a conservative sufficient condition. The orbital debris model states this explicitly through $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$3, and the general Wasserstein DRCCP literature treats CVaR as a convex inner approximation rather than an equivalence in the generic case (Ryu et al., 2024, Hota et al., 2018).

Computationally, exact shared DRCCs are often nonconvex and expensive. Wasserstein DR joint chance constraints can lead to mixed-integer linear programs or bilinear programs with size scaling like samples times network size times contingencies. Exact DRCC feasibility sets are strongly NP-hard even for affine uncertainty, and mixed-integer nonlinear formulations arise in generalized Nash games with shared DRCCs. These difficulties explain the prominence of CVaR surrogates, polyhedral approximations, reformulation via robust counterparts, and bisection-based methods such as ALSO-X# (Maghami et al., 2022, Hota et al., 2018, Jiang et al., 2023).

Conservatism is controlled by risk and ambiguity parameters, but the direction of the trade-off is uniform across domains. Smaller $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$4 in orbital collision avoidance produces stricter safety, larger separation, and higher fuel usage. Larger Wasserstein radius $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$5 in dispatch raises cost and lowers violation frequency. In power dispatch, a shared ambiguity set improves robustness to correlation uncertainty but can also increase conservatism; correlation-aware polyhedral sets reduce that conservatism relative to box sets (Ryu et al., 2024, Maghami et al., 2022).

Different ambiguity metrics behave very differently at low risk levels. For TVD, Hellinger, KL, and Neyman $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$6-distance, the perturbed risk level collapses rapidly as $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$7; for RVD,

$$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$8

so the ratio $$\inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)\le 0,\ i=1,\dots,m\big)\ge 1-\epsilon,$$9 remains constant. This makes RVD particularly attractive for shared DRCCs in MPC and randomized design when very small system-level violation probabilities are required (Heinlein et al., 2024).

Two unresolved issues recur. One is multistage structure: Bayesian DRCC work identifies multi-stage extensions as an open and practical issue. The other is scalable risk allocation: optimizing $$\sum_{i=1}^m \alpha_i\le \epsilon,\qquad \inf_{Q\in\mathcal{M}_N^\theta}Q\big(g_i(x,\xi)>0\big)\le \alpha_i,$$0 in shared-budget DRCCs is already present in Bayesian formulations, but its algorithmic treatment is incomplete in large-scale coupled systems (Chen et al., 2023).

A broader review perspective frames DRCCs as conservative approximations to classical chance constraints whenever the true distribution belongs to the ambiguity set. Within that perspective, shared DRCCs occupy the interface of chance-constrained optimization, robust optimization, and risk-averse modeling: they preserve a probabilistic semantics, but the probability guarantee is carried by a structured family of distributions rather than by a single estimated law (Rahimian et al., 2019).