Data-Driven Contextual Uncertainty Sets

• The paper presents a unified framework integrating statistical hypothesis tests and convex duality to construct data-driven uncertainty sets with finite-sample guarantees.
• It demonstrates flexibility by accommodating various tests and distributional assumptions, illustrated through applications in portfolio management and queueing systems.
• The approach ensures computational tractability via efficient reformulations, such as conic quadratic and LMI methods, while exploiting contextual information.

A data-driven contextual uncertainty set is a convex or non-convex set in robust optimization whose construction is directly informed by historical data and shaped by statistical properties identified through hypothesis testing, confidence regions, or empirical quantile estimators. This design exploits auxiliary information—contextual features, side-information, class labels, or correlated variables—to tailor the set to the operational scenario, thus enhancing the trade-off between conservativeness, tractability, and finite-sample guarantees (Bertsimas et al., 2013).

1. Construction via Statistical Hypothesis Testing

The central methodology begins by defining a confidence region for the unknown parameter distribution $P^*$ using statistical hypothesis tests. The confidence region %%%%1%%%% comprises those distributions not rejected at significance level $\alpha$ (e.g., Pearson’s χ²-test, Kolmogorov–Smirnov test). This confidence region is translated into an uncertainty set $\mathcal{U}(S, \varepsilon, \alpha)$ for the uncertain parameter vector $u$ such that, if $x$ is feasible for all $u \in \mathcal{U}$ in

$f(u, x) \leq 0, \qquad \forall u \in \mathcal{U}$

then with high probability (at least $1 - \alpha$) $x$ satisfies

$P^*(f(u, x) \leq 0) \geq 1 - \varepsilon$

for the original chance constraint.

For a concrete finite-support example, using Pearson’s χ²-test the confidence region is: $$\mathcal{P}^{\chi^2} = \left\{p \in \Delta_n : \sum_{i=0}^{n-1} \frac{(p_i - \hat{p}_i)^2}{2p_i} \leq \frac{\chi^2_{n-1, 1-\alpha}}{2N}\right\}$$ This confidence region is related—via convex duality—to a set $\mathcal{U}$ defined by the support function property: $$\delta^*(v \mid \mathcal{U}) \geq \text{VaR}_\varepsilon^{(P^*)}(v) \qquad \forall v \in \mathbb{R}^d$$ where $$\delta^*(v \mid \mathcal{U}) = \sup_{u \in \mathcal{U}} v^\top u$$, and $\text{VaR}$ is the Value-at-Risk over distributions in the confidence region.

The generic schema:

1. Test selection: Choose a statistical test (χ², KS, t, etc.) and define $\mathcal{P}(S, \alpha)$.
2. Worst-case quantile calculation: For $v \in \mathbb{R}^d$, compute the upper bound on worst-case $\text{VaR}_\varepsilon^{(P)}(v)$ for $P \in \mathcal{P}(S, \alpha)$.
3. Set construction: Via convexity and duality, define $\mathcal{U}(S, \varepsilon, \alpha)$ as the set whose support function matches the derived bound.

The resulting $\mathcal{U}$ adapts its geometry (e.g., polyhedral, ellipsoidal, union of intervals) based on both the statistical structure of the test and the nature of the contextual information present in the data.

2. Flexibility and Applicability

By varying both the nature of the statistical test and the structural assumptions about $P^*$—including exchangeability, independence, discreteness, or marginal knowledge—the schema supports multiple forms of uncertainty sets:

• Discrete support: Sets such as $\mathcal{U}_\varepsilon^{\chi^2}$ (using χ² or G-tests) and those based on marginal empirical distributions.
• Continuous support: Employing Kolmogorov–Smirnov or related tests for independent marginals or more complex dependence structures.
• Per-marginal or order statistic–driven sets: For asynchronously observed features or partial context.

In multi-constraint robust optimization problems, the schema allows the construction of constraint-specific sets—potentially, different $\varepsilon_j$ per constraint, yielding less conservative results in many practical applications.

Examples:

• Portfolio Management: Building robust portfolios by selecting an uncertainty set (e.g., $\hat{M}_\varepsilon$, $\mathcal{U}_\varepsilon^{LCX}$) to trade off conservatism and in-sample performance.
• Queueing: Calibrating waiting time bounds in G/G/1 queues with uncertainty sets derived from observed service/interarrival time residuals.

This flexibility makes the method directly applicable to operations research, supply chain management, and queuing analysis environments where contextual data is abundant.

3. Tractability and Optimization Reformulations

A distinguishing feature of the approach is the preservation of computational tractability:

• For functions $f(u,x)$ that are bi-affine, conic, or separable, robust counterparts are reformulated using the support function $\delta^*(v \mid \mathcal{U})$, often resulting in conic quadratic, second-order cone, or LMI formulations.

Explicit, efficiently computable support function representations are given for the uncertainty sets (e.g., $\hat{M}_\varepsilon$, sets based on moment constraints, order statistics, or convex ordering). In cases where the support function involves nonlinearities (such as relative entropy), efficient cutting-plane and separation algorithms are prescribed.

Key computational insights:

• The robust counterparts, even when built from high-dimensional or complex-valued data–driven sets, remain polynomial-time solvable by state-of-the-art conic solvers.
• Simultaneous satisfaction of multiple chance constraints is certified with finite-sample guarantees, and per-constraint “size” parameters can be efficiently tuned.

4. Probabilistic and Finite-Sample Guarantees

A central contribution of the framework is the finite-sample, data-dependent coverage guarantee: For any set $\mathcal{U}_\varepsilon$ constructed using a statistical test at significance $\alpha$, then with probability at least $1-\alpha$, all solutions robust to $\mathcal{U}_\varepsilon$ will satisfy

$P^*(f(u,x) \leq 0) \geq 1 - \varepsilon$

for any underlying $P^*$ meeting the assumed structure.

The guarantee fundamentally depends only on:

• The significance level $\alpha$ (chosen by the decision maker or regulator),
• Data quantity and quality, and
• Structural assumptions about $P^*$ (e.g., independence, support).

As the sample size increases, the uncertainty set shrinks (due to improved estimation accuracy), directly reducing conservativeness while maintaining the same probabilistic guarantee.

5. Numerical Performance and Comparison

Computational experiments—particularly in portfolio management and queueing—demonstrate:

• Data-driven uncertainty sets substantially reduce solution conservatism and improve empirical performance over classical non–data-informed robust approaches.
• In portfolio problems, moment- and dependence–aware sets (e.g., $\hat{M}_\varepsilon$, $\mathcal{U}_\varepsilon^{LCX}$) allow portfolios with higher realized returns for the same level of out-of-sample risk.
• In queueing, the data-driven approach outperforms traditional bounds (e.g., Kingman’s bound), delivering sharper and more stable waiting time estimates, as the uncertainty set appropriately shrinks with growing data.

Critical metrics include in-sample/out-of-sample gap, solution variability (across data samples), and the conservativeness (size) of the uncertainty set. Cross-validation and simulation studies in these domains empirically validate the value of using contextual historical data.

6. Theoretical and Methodological Impact

The methodological unification introduced by the schema combines elements of statistical hypothesis testing, convex duality, and robust chance-constrained programming. This bridges classical robust optimization with modern data-driven methods, providing:

• A rigorous route for translating observational data into set-based uncertainty representations,
• A systematic way to size uncertainty sets using coverage thresholds and data-driven insights,
• Decision-relevant finite-sample confidence guarantees under minimal distributional assumptions.

The approach generalizes earlier data-driven robust optimization frameworks and supplies a foundation for adapting robust optimization models as richer and larger datasets become available.

7. Outlook and Limitations

While the paper establishes a strong foundation for constructing tractable, probabilistically guaranteed uncertainty sets from contextual data, several boundaries remain:

• The framework’s performance is tied to the representativeness of the historical (contextual) data and the validity of structural assumptions for $P^*$.
• Nonconvexities (e.g., when employing worst-case quantiles in nonconvex support) are addressed via convex surrogates (CVaR upper bounds), guaranteeing tractability but potentially introducing additional conservatism.

This approach is directly extensible—combining with other flexible machine learning or context-sensitive distribution learning methods—opening research directions for more sophisticated contextual or adaptive set constructions. The finite-sample guarantees and explicit reformulations ensure ongoing practical significance in robust operational decision-making (Bertsimas et al., 2013).