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Decision-Scaled Scenario Approach for Rare Chance-Constrained Optimization

Jaeseok Choi, Anand Deo, Constantino Lagoa, Anirudh Subramanyam

Abstract

Chance-constrained optimization is a suitable modeling framework for safety-critical applications where violating constraints is nearly unacceptable. The scenario approach is a popular solution method for these problems, due to its straightforward implementation and ability to preserve problem structure. However, in the rare-event regime where constraint violations must be kept extremely unlikely, the scenario approach becomes computationally infeasible due to the excessively large sample sizes it demands. We address this limitation with a new yet straightforward decision-scaling method that relies exclusively on original data samples and a single scalar hyperparameter that scales the constraints in a way amenable to standard solvers. Our method leverages large deviation principles under mild nonparametric assumptions satisfied by commonly used distribution families in practice. For a broad class of problems satisfying certain practically verifiable structural assumptions, the method achieves a polynomial reduction in sample size requirements compared to the classical scenario approach, while also guaranteeing asymptotic feasibility in the rare-event regime. Numerical experiments spanning finance and engineering applications show that our decision-scaling method significantly expands the scope of problems that can be solved both efficiently and reliably.

Decision-Scaled Scenario Approach for Rare Chance-Constrained Optimization

Abstract

Chance-constrained optimization is a suitable modeling framework for safety-critical applications where violating constraints is nearly unacceptable. The scenario approach is a popular solution method for these problems, due to its straightforward implementation and ability to preserve problem structure. However, in the rare-event regime where constraint violations must be kept extremely unlikely, the scenario approach becomes computationally infeasible due to the excessively large sample sizes it demands. We address this limitation with a new yet straightforward decision-scaling method that relies exclusively on original data samples and a single scalar hyperparameter that scales the constraints in a way amenable to standard solvers. Our method leverages large deviation principles under mild nonparametric assumptions satisfied by commonly used distribution families in practice. For a broad class of problems satisfying certain practically verifiable structural assumptions, the method achieves a polynomial reduction in sample size requirements compared to the classical scenario approach, while also guaranteeing asymptotic feasibility in the rare-event regime. Numerical experiments spanning finance and engineering applications show that our decision-scaling method significantly expands the scope of problems that can be solved both efficiently and reliably.
Paper Structure (33 sections, 18 theorems, 169 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 33 sections, 18 theorems, 169 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Outline.
  4. Notations.
  5. Preliminaries and Assumptions
  6. Regular Variation
  7. Assumptions on the Uncertainty Distribution
  8. Assumptions on the Constraint Function
  9. Decision-Scaled sa
  10. Overview of Classical sa
  11. Decision-scaled sa
  12. Verification of Regularity Conditions and Examples
  13. Verification of Assumption \ref{['assum:conti-conv']} for Algebraic Functions
  14. Illustrative Examples.
  15. Extension via Constraint Decomposition
  16. ...and 18 more sections

Key Result

Proposition 2.7

Under Assumptions Eq:Conti-ConvergenceAssum:NonEmptyHorizoneq:away_from_0eq:nonempty_set_g^*, the pair $(\gamma,\rho)$ is unique.

Figures (6)

  • Figure 1: Comparison of computational efficiency for the portfolio optimization problem: (left) Required sample size; (right) CPU time.
  • Figure 2: Comparison of solution quality for the portfolio optimization problem: (left) Objective value; (right) Violation probability.
  • Figure 3: Comparison of computational efficiency for the short column design problem: (left) Required sample size; (right) CPU time.
  • Figure 4: Comparison of solution quality for the short column problem: (left) Objective value; (right) Violation probability.
  • Figure 5: Comparison of computational efficiency for the norm optimization problem: (left) Required sample size; (right) CPU time. At $\varepsilon=10^{-5}$, the classical sa failed to find a feasible solution for any of the 100 test instances.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.5: Asymptotic Cone
  • Proposition 2.7: Uniqueness of Scaling
  • Definition 3.1
  • Theorem 3.2: campi2009scenario, Theorem 1
  • Theorem 3.3
  • Corollary 3.4
  • Remark 4.1
  • ...and 40 more