Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sample Complexity of Chance Constrained Optimization in Dynamic Environment

Apurv Shukla, Qian Zhang, Le Xie

TL;DR

This work extends the scenario approach for chance-constrained optimization to dynamic environments with time-varying, potentially drifting distributions. By coupling successive scenario-generating measures with the 1-Wasserstein distance, it derives ex-post risk guarantees for both convex and non-convex feasible sets, recovering classical static results as drift vanishes. The main contributions include explicit sample-complexity bounds that quantify how non-stationarity affects required scenarios, and validation through convex and non-convex numerical experiments such as a probabilistic point-covering problem and a mixed-integer control task. The results provide a principled framework for robust decision-making under evolving uncertainty, with practical implications for energy systems and other time-varying domains.

Abstract

We study the scenario approach for solving chance-constrained optimization in time-coupled dynamic environments. Scenario generation methods approximate the true feasible region from scenarios generated independently and identically from the actual distribution. In this paper, we consider this problem in a dynamic environment, where the scenarios are assumed to be drawn sequentially from an unknown and time-varying distribution. Such dynamic environments are driven by changing environmental conditions that could be found in many real-world applications such as energy systems. We couple the time-varying distributions using the Wasserstein metric between the sequence of scenario-generating distributions and the actual chance-constrained distribution. Our main results are bounds on the number of samples essential for ensuring the ex-post risk in chance-constrained optimization problems when the underlying feasible set is convex or non-convex. Finally, our results are illustrated on multiple numerical experiments for both types of feasible sets.

Sample Complexity of Chance Constrained Optimization in Dynamic Environment

TL;DR

This work extends the scenario approach for chance-constrained optimization to dynamic environments with time-varying, potentially drifting distributions. By coupling successive scenario-generating measures with the 1-Wasserstein distance, it derives ex-post risk guarantees for both convex and non-convex feasible sets, recovering classical static results as drift vanishes. The main contributions include explicit sample-complexity bounds that quantify how non-stationarity affects required scenarios, and validation through convex and non-convex numerical experiments such as a probabilistic point-covering problem and a mixed-integer control task. The results provide a principled framework for robust decision-making under evolving uncertainty, with practical implications for energy systems and other time-varying domains.

Abstract

We study the scenario approach for solving chance-constrained optimization in time-coupled dynamic environments. Scenario generation methods approximate the true feasible region from scenarios generated independently and identically from the actual distribution. In this paper, we consider this problem in a dynamic environment, where the scenarios are assumed to be drawn sequentially from an unknown and time-varying distribution. Such dynamic environments are driven by changing environmental conditions that could be found in many real-world applications such as energy systems. We couple the time-varying distributions using the Wasserstein metric between the sequence of scenario-generating distributions and the actual chance-constrained distribution. Our main results are bounds on the number of samples essential for ensuring the ex-post risk in chance-constrained optimization problems when the underlying feasible set is convex or non-convex. Finally, our results are illustrated on multiple numerical experiments for both types of feasible sets.
Paper Structure (13 sections, 9 theorems, 44 equations, 4 figures, 1 table)

This paper contains 13 sections, 9 theorems, 44 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Chance Constrained Optimization with Observation Robustness
  4. Chance-Constrained Optimization (CCO)
  5. Scenario Approach in Dynamic Environments
  6. Convex Scenario Problems
  7. Non-convex Scenario Problems
  8. Numerical Experiments
  9. Convex Feasible Region: The Probabilistic Point Covering Problem
  10. Non-convex Feasible region: Control with Quantized Inputs
  11. Conclusion
  12. The Analytical Expression of 1-Wasserstein Distance Between Two Normal Distribution
  13. Simulation Details

Key Result

Proposition 1

Under Assumption 1, the optimal solution to the scenario problem $\mathrm{SP}(\mathcal{S})$ satisfies: where, the probability $\mathbb{P}^N$ is taken with respect to $\textit{N}^{th}$ scenario, and h is the Helly's dimension of $\mathrm{SP}(\mathcal{S})$.

Figures (4)

  • Figure 1: The relationship between the measurement error term and confidence parameter
  • Figure 2: Comparing risk function between different $\rho$ and $r_0$ for mixed-integer optimal control
  • Figure 3: The approximation of 1-Wasserstein distance between $\textit{Norm}(\mu_i,\sigma_i)$ and $\textit{Norm}(\mu_{310},\sigma_{310})$
  • Figure 4: The changing distribution of $\xi_i$

Theorems & Definitions (20)

  • Definition 1: Violation Probability
  • Definition 2: Support Scenario
  • Definition 3: Helly's dimension
  • Proposition 1: Violation Probability calafiore2006scenario
  • Proposition 2
  • Definition 4: Violation in dynamic environments
  • Definition 5: Wasserstein metric villani2021topics
  • Definition 6: Model A
  • Definition 7: Model B
  • Lemma 1
  • ...and 10 more