Wasserstein Robust Joint Chance Constraints

• WJCC is a robust optimization framework that guarantees joint chance constraints hold across Wasserstein ambiguity sets, mitigating distributional uncertainty.
• It transforms probabilistic constraints into tractable deterministic models using exact reformulations and convex inner approximations such as CVaR and ALSO-X.
• Real-world applications in power systems, finance, and control validate its effectiveness in managing uncertainty while enhancing decision reliability.

Wasserstein Distributionally Robust Joint Chance Constraints (WJCC) formalize a class of optimization problems where the decision maker seeks to ensure that multiple constraints are satisfied simultaneously with high probability, uniformly across a set of probability distributions that are close (in the Wasserstein metric sense) to a nominal or empirical distribution. This framework extends classical (single-distribution) joint chance constraints by introducing an ambiguity set to hedge against distributional uncertainty, thereby providing probabilistic guarantees with respect to unknown or misspecified distributions. The field has seen significant developments in exact and approximate reformulations, scalable algorithms, convergence theory, and a growing body of real-world applications in areas such as power systems, finance, and control.

1. Problem Setting and Ambiguity Model

A classical joint chance constraint (JCC) requires that a family of constraints $f_j(x, \xi) \leq 0$ for $j=1,\ldots,s$ hold jointly with probability at least $1-\varepsilon$ under a known distribution $P$ for the uncertainty $\xi$: $$P\big( f_j(x, \xi) \leq 0 \; \forall j \big) \geq 1-\varepsilon.$$ Wasserstein Distributionally Robust Joint Chance Constraints generalize this by introducing an ambiguity set $\mathcal{P}$ comprising all probability measures within a prescribed Wasserstein distance $r$ from a nominal distribution $\hat{P}$, usually the empirical distribution from finitely many data samples. The WJCC requires

$$\inf_{Q \in \mathcal{P}} Q\big( f_j(x, \xi) \leq 0 \; \forall j \big) \geq 1 - \varepsilon.$$

This construction ensures joint constraint satisfaction for all distributions in the ambiguity set, providing robustness against misspecification and finite-sample uncertainty (Hota et al., 2018).

The Wasserstein ball is typically defined as

$$\mathcal{P} := \{Q \in \mathcal{P}(\Xi) : W_p(Q, \hat{P}) \leq r\},$$

where $W_p$ is the order-$p$ Wasserstein distance, $\hat{P}$ is the empirical (or nominal) measure, and $r$ tunes the desired level of robustness reflecting sampling error and confidence (Cherukuri et al., 2020, Gu et al., 2021).

2. Exact Reformulations and Computational Challenges

Exact reformulations for WJCC depend on the structure of the constraint functions $f_j$ and the choice of Wasserstein metric and norm.

• For joint chance constraints with right-hand side uncertainty and polyhedral safety sets, deterministic reformulations can be derived using optimal transport duality. The problem reduces to enforcing that the average cost of transporting the $\varepsilon N$ closest empirical samples to the unsafe set meets or exceeds the radius $r$; this leads to mixed-integer conic programs, or with specific norms, to mixed-integer linear programs (Chen et al., 2018). The core deterministic reformulation is:

  $$\frac{1}{N} \sum_{i=1}^{\varepsilon N} \mathrm{dist}(\xi_{\pi_i(x)}, \mathcal{C}(x)) \geq r,$$

  with $\{\xi_{\pi_i(x)}\}_{i=1}^{N}$ sorted by distance to the unsafe set.

• For general convex constraints or left-hand side uncertainty (decision-dependent uncertainty), strong reformulations are more difficult, often leading to nonconvex or high-dimensional mixed-integer programs (Gu et al., 2021, Zhang, 2023).
• In high dimensions or when the number of constraints grows, these exact reformulations rapidly become computationally intractable, and the presence of nonconvexity or binary variables is a bottleneck (Chen et al., 2018, Zhang, 2023). Special structures—such as one-dimensionality in constraints—can be exploited for further reformulation and acceleration (Zhou et al., 23 Jun 2025).

3. Inner Approximations and Convexification

Due to the computational challenges of exact formulations, several systematic convex inner approximation techniques have been developed:

• Worst-case Conditional Value-at-Risk (CVaR) Approach:

  The chance constraint is conservatively approximated by requiring that the worst-case CVaR (over the ambiguity set) is bounded:

  $$\sup_{Q \in \mathcal{P}} \mathrm{CVaR}_\varepsilon\left(\max_j \{f_j(x, \xi)\}\right) \leq 0.$$

  Tractable reformulations are available when $f_j$ is affine in both $x$ and $\xi$ (Hota et al., 2018, Gu et al., 2021, Chen et al., 2022).

  The CVaR approximation is equivalent to requiring that the transportation cost of moving the $\varepsilon$-fraction of most violating samples to the unsafe set is at least the Wasserstein ball's radius (Chen et al., 2018, Chen et al., 2022):

  $$\frac{1}{N} \sum_{i=1}^{\varepsilon N} \mathrm{dist}\bigl(\xi_{\pi_i(x)}, \overline{\mathcal{S}}(x)\bigr) \geq r.$$

  Moreover, for affine and binary settings, mixed-integer convex formulations and McCormick linearizations can be used to achieve scalability (Gu et al., 2021).

• Bonferroni and Scenario-Based Approximations:

  A union bound is used to decouple the joint constraint into individual ones, each treated at a risk level $\varepsilon_j$, with $\sum_j \varepsilon_j = \varepsilon$. This leads to tractable individual constraints but may be overly conservative, especially when the constraint functions are correlated (Chen et al., 2022).

• ALSO-X Family of Approximations:

  Recent work generalizes the CVaR and scenario approaches. The ALSO-X and ALSO-X# (Jiang et al., 2023) methods formulate a bilevel program, combining a refined lower-level CVaR-based loss minimization with an upper-level constraint satisfaction check. ALSO-X# can outperform classical CVaR or ALSO-X for convex and certain discrete (e.g., binary) feasible sets, and sufficient conditions are provided for exactness.

• Exploiting Problem Structure (FICA, SFLA):

  For WJCCs with one-dimensional or partially one-dimensional structure (such as those arising in power dispatch with left-hand or right-hand side uncertainties), methods such as FICA (Zhou et al., 23 Jun 2025) and Strengthened and Faster Linear Approximation (SFLA) (Zhou et al., 17 Dec 2024) migrate valid inequalities from exact formulations into a convex inner approximation. This substantially reduces the number of constraints and ancillary variables, achieving substantial computational speedup (up to $40\times$ or more compared to CVaR) while maintaining near-identical feasibility regions.

4. Consistency and Asymptotic Guarantees

One of the foundational properties of Wasserstein-based WJCC is asymptotic consistency: as the number of data samples increases and the ambiguity radius decreases appropriately, the set of robustly feasible decisions converges (in a suitable sense, such as Lebesgue measure) to the set defined under the true (unknown) distribution (Cherukuri et al., 2020, Lasserre et al., 2018). Formally, for a suitable sequence of radii $\{\theta_N\}$ tending to zero,

$$\lim_{N \to \infty} \lambda\big( X^*_{\varepsilon} \setminus X^d_{\varepsilon} \big) = 0,$$

where $X^*_{\varepsilon}$ is the true feasible region and $X^d_{\varepsilon}$ is the inner approximation at relaxation order $d$ (Lasserre et al., 2018). For Wasserstein ambiguity sets constructed with statistically valid radii, the robust optimizer converges to the optimizer of the population-level chance-constrained program, and finite-sample confidence guarantees are available (Cherukuri et al., 2020).

5. Algorithms and Computational Strategies

• SDP Hierarchies and Polynomial Approximations:

  When uncertainty is polynomial and ambiguity is given by mixtures with polynomial moments, sequences of inner approximations are computed via semidefinite programming (SDP), converging asymptotically to the feasible region of the joint chance constraint. This is robust to nonconvexity and can be implemented with standard moment-sum-of-squares packages, but scales primarily to moderate-sized problems (Lasserre et al., 2018).

• Cutting-Plane Algorithms:

  For problems with semi-infinite robustified constraints (e.g., concave or convex $f_j$ in $\xi$), central cutting-plane methods iteratively add violated constraints (cuts) and solve a master convex program until an $\eta$-approximate solution is found (Hota et al., 2018).

• Mixed-Integer Linear and Conic Programs:

  For problems with right-hand side uncertainty and under suitable norm choices, the deterministic reformulation produces mixed-integer linear programs (MILPs) whose size grows linearly with the number of empirical samples and constraints, leading to tractable solutions with commercial solvers for moderately large instances (Chen et al., 2018, Zhou et al., 17 Dec 2024, Zhang, 2023).

• Two-Step Approximation Approaches:

  For scalable robustification in large-scale power dispatch or security-constrained optimization, approaches split the problem into (i) constructing a polyhedral uncertainty set capturing at least $1-\varepsilon$ mass for all distributions in the ambiguity set, and (ii) enforcing robust constraints over this set using duality and standard robust optimization techniques, dramatically improving scalability (Maghami et al., 2022).

• Exploiting Structure with Valid Cuts:

  For set covering, knapsack, and combinatorial settings with discrete uncertainty and/or binary decisions, the two-stage model can be enhanced with valid inequalities—such as extended polymatroid inequalities—that sharply strengthen the master problem and speed up decomposition-based algorithms (Shen et al., 2020, Zhang, 2023).

• Bayesian Data-Driven Uncertainty Sets:

  Integration of Bayesian credible intervals for parameter uncertainty provides universal and flexible ambiguity set construction, allowing tractable robust counterparts with explicit finite-sample guarantees and the ability to encode prior information (Chen et al., 2023).

6. Applications and Impact

Wasserstein Distributionally Robust Joint Chance Constraint frameworks have seen diverse applications:

• Energy Systems and Power Markets:

  Unit commitment and grid dispatch problems account for wind and solar uncertainty in constraints on line flows and dispatch levels, seeking joint reliability guarantees across multiple constraints (Zhou et al., 17 Dec 2024, Zhou et al., 23 Jun 2025, Wang et al., 2021, Maghami et al., 2022). Approaches such as SFLA and FICA have been shown to yield $10\times$ to $500\times$ computational improvement over previous methods.

• Model Predictive Control (MPC):

  In stochastic MPC, joint state and input constraints are enforced distributionally robustly with respect to Wasserstein sets. Support from tractable second-order cone or convex programs ensures recursive feasibility and high reliability even under finite data (Mark et al., 2020, Zhong et al., 2021).

• Facility Location and Humanitarian Logistics:

  Multi-period location and capacity problems ensure that all service constraints are met with high probability under uncertain, dynamically evolving demand, using $\infty$-Wasserstein balls for ambiguity modeling (Wang et al., 2021).

• Portfolio Optimization and Finance:

  Portfolio policies constructed via Wasserstein-robust joint chance constraints offer out-of-sample reliability, balancing risk and return under ambiguous asset return distributions (Shen et al., 2022, Chen et al., 2023).

• Wireless Communication and Resource Allocation:

  Advanced convex approximations (ALSO-X#) enable near-exact and efficient handling of robust chance constraints with integer or combinatorial decisions (Jiang et al., 2023).

7. Approximation-Quality, Trade-offs, and Open Directions

The theoretical and empirical analyses reveal that:

• CVaR-based approximations are typically tight convex inner approximations but can be conservative, especially in high dimension or for correlated constraint violations (Chen et al., 2022, Gu et al., 2021).
• Bonferroni-type decompositions are computationally simpler but may be substantially more conservative as the number of joint constraints grows.
• The ALSO-X and ALSO-X# approaches interpolate between CVaR and scenario, sometimes achieving better approximation quality, particularly for discrete feasible sets (Jiang et al., 2023).
• Structure-exploiting methods (such as FICA and SFLA) provide substantial computational speedups without sacrificing reliability where their conditions apply (Zhou et al., 17 Dec 2024, Zhou et al., 23 Jun 2025).
• For all approximation methods, tuning the Wasserstein radius is critical for controlling out-of-sample feasibility and balancing robustness against sample-size-induced conservatism (Cherukuri et al., 2020, Shen et al., 2022, Zhong et al., 2021).
• In data-driven practice, no single approximation is uniformly best; trade-offs must be managed based on dimensionality, constraint structure, computational resources, and tolerance for conservatism (Chen et al., 2022).

Open questions include extending scalable approaches to general (nonlinear, high-dimensional) left-hand side uncertainties, principled hyperparameter (e.g., $\kappa_i$) tuning, and integration with scenario reduction and Bayesian/DRO hybrid frameworks.

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Table: Summary of Key Methodological Approaches in WJCC

Approach	Formulation	Scalability
Exact Reformulation (MIP)	Mixed-integer conic/linear programs (Chen et al., 2018, Zhang, 2023, Gu et al., 2021)	Moderate
CVaR Inner Approximation	Convex (conic) programs (Hota et al., 2018, Chen et al., 2022, Gu et al., 2021)	Good (for affine/convex)
Bonferroni/Union Bound	Decoupled individual constraints (Chen et al., 2022)	High, but conservative
ALSO-X / ALSO-X#	Bilevel/convex approximations (Jiang et al., 2023)	High, improves on CVaR in some cases
Structure-exploiting (FICA, SFLA)	Fast convex approx. for special structure (Zhou et al., 23 Jun 2025, Zhou et al., 17 Dec 2024)	Excellent for structured cases
Polyhedral uncertainty set	Two-step RO (Maghami et al., 2022)	High (large-scale systems)

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References

Detailed results and methodologies can be found in (Lasserre et al., 2018, Hota et al., 2018, Chen et al., 2018, Cherukuri et al., 2020, Gu et al., 2021, Zhong et al., 2021, Wang et al., 2021, Shen et al., 2022, Chen et al., 2022, Maghami et al., 2022, Jiang et al., 2023, Zhang, 2023, Chen et al., 2023, Zhou et al., 17 Dec 2024), and (Zhou et al., 23 Jun 2025).