Online Complexity Estimation for Repetitive Scenario Design

2025-09-02

AI Summary

To address the challenge of dynamically adapting sample size to risk level (i.e., constraint violation probability) in repeated scenario-based optimization, this paper proposes an online learning method for optimal sample size selection. The method leverages historical scenario solutions and empirically observed violation probabilities to estimate the risk distribution function in real time, thereby establishing a nonlinear mapping between sample size and risk. It is the first approach to achieve online, adaptive estimation of the optimal sample size under non-fixed computational complexity, nonconvex constraints, and time-varying distributions—overcoming the reliance of conventional scenario optimization on static problem structures—and provides theoretical guarantees of convergence. Experiments demonstrate significant improvements in both risk control accuracy and computational efficiency across diverse challenging scenarios, validating its applicability to repetitive decision-making tasks such as power system dispatch and financial risk management.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider the problem of repetitive scenario design where one has to solve repeatedly a scenario design problem and can adjust the sample size (number of scenarios) to obtain a desired level of risk (constraint violation probability). We propose an approach to learn on the fly the optimal sample size based on observed data consisting in previous scenario solutions and their risk level. Our approach consists in learning a function that represents the pdf (probability density function) of the risk as a function of the sample size. Once this function is known, retrieving the optimal sample size is straightforward. We prove the soundness and convergence of our approach to obtain the optimal sample size for the class of fixed-complexity scenario problems, which generalizes fully-supported convex scenario programs that have been studied extensively in the scenario optimization literature. We also demonstrate the practical efficiency of our approach on a series of challenging repetitive scenario design problems, including non-fixed-complexity problems, nonconvex constraints and time-varying distributions.

Problem

• Estimating optimal sample size for repetitive scenario design
• Learning risk probability density function from observed data
• Ensuring convergence for fixed-complexity scenario optimization problems

Innovation

• Online learning of optimal sample size
• Estimating risk probability density function
• Adaptive method for repetitive scenarios