[Paper Review] Solving contextual chance-constrained programming under decision-dependent uncertainty

Xiangting Liu, Shengran Wang+2 more|arXiv (Cornell University)|Feb 7, 2026

Risk and Portfolio Optimization0 citations

TL;DR

This paper introduces Contextual Cluster Weights (CCW), a non-parametric framework to solve contextual chance-constrained programming when decisions affect uncertainty, with uniform-in-decision guarantees and a scalable solution via reformulations and clustering.

ABSTRACT

We study contextual chance-constrained programming under decision-dependent uncertainty. In this setting, a decision not only needs to satisfy constraints but also alters the distribution of uncertain outcomes. This dependency makes the problem particularly difficult: because feasibility probabilities vary with decisions, it creates both statistical endogeneity and computational intractability. To address this, we propose a nonparametric approximation method based on Contextual Cluster Weights (CCW). For any given decision and context, CCW constructs a local neighborhood (cluster) of ``similar" historical observations and assigns them equal weight. This approach successfully renders both the objective and chance constraints tractable, while providing uniform-in-decision consistency guarantees. Furthermore, we develop reformulations that use pre-calculated clusters. We show that under a specific nestedness condition, these reformulations yield a convex feasible region, which allows for efficient solving. Experiments, including a case study with JD.com, demonstrate that our method outperforms benchmarks in solution quality, feasibility reliability, and runtime. This framework offers a scalable and data-driven approach for firms to make reliable operational decisions when their actions influence uncertainty. It effectively balances performance, risk, and robustness, while remaining interpretable and implementable in practice.

Motivation & Objective

• Address the challenge of contextual chance-constrained programming when decisions influence the distribution of uncertain outcomes (decision-dependent uncertainty).
• Develop a non-parametric CCW approach to approximate objectives and chance constraints with uniform-in-decision consistency.
• Provide reformulations that leverage pre-calculated clusters to achieve computational tractability and convex feasible regions under a nestedness condition.
• Demonstrate theoretical guarantees and empirical performance on synthetic tests and a JD.com case study.

Proposed method

• Formulate CCCP-DDU and propose a non-parametric CCW approximation where weights are assigned based on local clusters of similar decision-context observations.
• Define CCW as set-based weights that assign equal weight to points in a local neighborhood and zero elsewhere (w_i(z,x) = I{(z_i,x_i) in C(z,x)}/|C(z,x)|).
• Use CCW to approximate the objective L(z|X=x) and the chance constraints g_xi(z|X=x) with uniformly convergent estimators (L-hat and g-hat).
• Provide specific CCW weight choices (kNN, CART, LSA) with corresponding consistency results and convergence rates.
• Show that under a nestedness condition, the reformulated problem yields a convex feasible region enabling efficient solving (e.g., via Benders decomposition).
• Model the approximate problem Appr-dd-ccp as a MINLP (21a–21e) that selects cluster members and enforces the chance constraints within the cluster.

Experimental results

Research questions

• RQ1 How can we reliably approximate context-conditioned objective and feasibility probabilities when decisions influence uncertainty distributions?
• RQ2 Can a non-parametric, cluster-based weighting scheme provide uniform-in-decision consistency for CCCP with decision-dependent uncertainty?
• RQ3 Under what conditions can the resulting reformulated problem be solved efficiently (e.g., convexity or decomposability)?
• RQ4 Do CCW-based methods outperform parametric benchmarks and existing non-parametric methods in terms of solution quality, feasibility reliability, and runtime?
• RQ5 What is the empirical impact of CCW in a real-world setting such as JD.com transaction data?

Key findings

• CCW provides a uniform-in-decision consistent non-parametric estimator for both the objective and the chance constraints under decision-dependent uncertainty.
• kNN and LSA CCW schemes yield strong uniform consistency for the probabilistic constraints and sample-efficiency rates; CART yields weak uniform consistency.
• Under a nestedness condition, the CCW-based reformulation yields a convex feasible region, enabling efficient optimization (e.g., via Benders decomposition).
• The MINLP (21a–21e) captures cluster selection and constraint satisfaction within a cluster, enabling tractable solving of the approximate CCCP-DDU.
• Empirical results on JD.com data show the CCW approach outperforms benchmarks in solution quality, feasibility reliability, and runtime.