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Solving contextual chance-constrained programming under decision-dependent uncertainty

Xiangting Liu, Shengran Wang, Kaile Yan, Zhi-Hai Zhang

TL;DR

The paper tackles contextual chance-constrained programming with decision-dependent uncertainty (DDU), where decisions alter the distribution of uncertain outcomes and feasibility probabilities. It introduces Contextual Cluster Weights (CCW), a nonparametric, set-based weighting scheme that forms local decision-context clusters to render both objective and chance constraints tractable while delivering uniform-in-decision consistency. A linearization reformulation coupled with a nested-structure convexity condition enables efficient solution via Benders decomposition, transforming a challenging MINLP into solvable MILP/subproblems. Empirical evaluation on PSNP and a JD.com case demonstrates superior solution quality, feasibility reliability, and runtime performance compared with parametric and existing nonparametric benchmarks, illustrating scalable, data-driven decision-making under endogenous uncertainty.

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.

Solving contextual chance-constrained programming under decision-dependent uncertainty

TL;DR

The paper tackles contextual chance-constrained programming with decision-dependent uncertainty (DDU), where decisions alter the distribution of uncertain outcomes and feasibility probabilities. It introduces Contextual Cluster Weights (CCW), a nonparametric, set-based weighting scheme that forms local decision-context clusters to render both objective and chance constraints tractable while delivering uniform-in-decision consistency. A linearization reformulation coupled with a nested-structure convexity condition enables efficient solution via Benders decomposition, transforming a challenging MINLP into solvable MILP/subproblems. Empirical evaluation on PSNP and a JD.com case demonstrates superior solution quality, feasibility reliability, and runtime performance compared with parametric and existing nonparametric benchmarks, illustrating scalable, data-driven decision-making under endogenous uncertainty.

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.
Paper Structure (43 sections, 14 theorems, 88 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 43 sections, 14 theorems, 88 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Relevant Literature
  3. CSO with Chance Constraints (CCCP)
  4. CSO with Decision-dependent Uncertainty
  5. Preliminaries
  6. Asymptotic Results
  7. Assumptions
  8. Asymptotic consistency
  9. Solution Framework
  10. Linearization Reformulation of the Model
  11. Special Case: Convexity Certification via Nested Structure
  12. Illustrative Example: Price-setting Newsvendor Problem
  13. Reformulation of the Approximated Chance Constraint
  14. Structural Analysis and Efficient Solution
  15. Numerical Experiment
  16. ...and 28 more sections

Key Result

Lemma 1

Suppose Assumptions ass::continuous, ass::regularity and ass::boundedtail hold. For some $0<C<1, 0.5<\delta_{kNN}<1$, let $k=\lceil CN^{\delta_{kNN}}\rceil$, then $\hat{L}(z\mid X=x)$ based on eq-kNN weight has strongly uniform consistency.

Figures (10)

  • Figure 1: Visual illustration of the nested property.
  • Figure 2: A demonstration of the loss function and the VaR constraint with different $d$.
  • Figure 3: Sample Efficiency under Different Relationship and Uncertainty Modes.
  • Figure 4: Verification of sample efficiency
  • Figure 5: Comparison of the computation time of four solution strategies.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Definition 1: kNN weight
  • Definition 2: CART weight
  • Definition 3: LSA weight
  • Definition 4: Uniform-in-decision Consistency
  • Lemma 1: kNN consistency
  • Lemma 2: CART consistency
  • Lemma 3: LSA consistency
  • Proposition 1: Consistency of the denominator and numerator
  • Proposition 2: Consistency of the fraction
  • Theorem 1: Consistency of estimating chance constraint with CCW
  • ...and 11 more