Guaranteeing Risk Levels in Chance-Constrained Optimization under Time-Varying Distributions

핵심 개념

The core message of this paper is to establish bounds on the violation probability of an optimal solution of the robust scenario problem for guaranteeing prescribed risk levels in chance-constrained optimization when the scenarios are generated from time-varying distributions, for both convex and non-convex feasible regions.

초록

The paper studies the scenario approach for solving chance-constrained optimization in time-coupled dynamic environments. The scenarios are assumed to be drawn in a sequential fashion from an unknown and time-varying distribution. The authors couple the time-varying distributions using the Wasserstein metric and provide 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. For the convex case, the authors show that the violation probability of the optimal solution to the robust scenario problem is bounded by a function of the Helly's dimension of the problem, the number of scenarios, and the Wasserstein distance between the scenario-generating distributions. For the non-convex case, the authors define an "invariant set" of constraints and provide bounds on the violation probability in terms of the cardinality of this set. The results are illustrated through numerical experiments on a probabilistic point covering problem (convex) and a mixed-integer optimal control problem (non-convex). The experiments demonstrate the tightness of the theoretical guarantees and show that they gracefully scale with the non-stationarity of the scenario-generating distribution.

통계

The number of scenarios N is used to bound the violation probability. The Helly's dimension h of the scenario problem is used to bound the violation probability in the convex case. The cardinality of the invariant set of constraints is used to bound the violation probability in the non-convex case. The Wasserstein distance ρ between the scenario-generating distributions is used to quantify the non-stationarity of the environment.

인용구

"The core message of this paper is to establish bounds on the violation probability of an optimal solution of the robust scenario problem for guaranteeing prescribed risk levels in chance-constrained optimization when the scenarios are generated from time-varying distributions, for both convex and non-convex feasible regions." "The authors couple the time-varying distributions using the Wasserstein metric and provide 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."

핵심 통찰 요약

Sample Complexity of Chance Constrained Optimization in Dynamic Environment

by Apurv Shukla... 게시일 arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00608.pdf

더 깊은 질문

How can the proposed approach be extended to handle more complex time-varying distributions, such as those with non-Gaussian or heavy-tailed characteristics

The proposed approach can be extended to handle more complex time-varying distributions by incorporating techniques from non-parametric statistics and machine learning. For non-Gaussian distributions, methods like kernel density estimation can be used to estimate the underlying distribution of the scenarios. By representing the distributions in a high-dimensional feature space, the Wasserstein metric can still be applied to measure the distance between distributions. Additionally, techniques like Gaussian mixture models or copulas can be employed to capture dependencies and non-Gaussian characteristics in the distributions. For heavy-tailed distributions, robust estimation methods can be utilized to account for outliers and extreme values in the scenario generation process. By adapting the scenario approach to handle these more complex distributions, the sample complexity bounds can be extended to ensure robustness and accuracy in decision-making under uncertainty.

What are the implications of the non-convex results on the design of robust control systems in the presence of uncertainty and time-varying disturbances

The implications of the non-convex results on the design of robust control systems in the presence of uncertainty and time-varying disturbances are significant. Non-convexity introduces challenges in optimization due to the presence of multiple local optima and complex feasible regions. In the context of robust control systems, non-convexity can lead to more intricate decision-making processes where traditional convex optimization techniques may not be directly applicable. The results from this work provide insights into how to navigate these challenges by considering the cardinality of invariant sets and developing strategies to ensure feasibility and robustness in non-convex scenarios. By leveraging the sample complexity bounds and the concept of invariant constraints, designers of robust control systems can tailor their approaches to handle the complexities introduced by non-convexity, leading to more resilient and adaptive control strategies in dynamic environments.

Can the insights from this work be leveraged to develop online or adaptive sampling strategies that can dynamically adjust the number of scenarios as the environment evolves over time

The insights from this work can be leveraged to develop online or adaptive sampling strategies that dynamically adjust the number of scenarios as the environment evolves over time. By incorporating real-time data and feedback mechanisms, adaptive sampling strategies can continuously update the scenario generation process based on the changing distribution of uncertainties. Techniques like reinforcement learning and online optimization can be employed to adjust the sampling strategy in response to new information and evolving environmental conditions. This adaptive approach can help in optimizing the trade-off between computational complexity and decision accuracy, ensuring that the scenario generation process remains effective and efficient in dynamic environments. By integrating the principles of adaptive sampling with the robust scenario approach, decision-makers can enhance the responsiveness and adaptability of their systems to changing conditions.