标题： 数据驱动的参数优化问题性能保证 显示英文标题

标题： Data-Driven Performance Guarantees for Parametric Optimization Problems

摘要： 我们提出了一种数据驱动的方法，以建立通过迭代算法求解的参数优化问题的概率性能保证。 我们的方法解决了两个关键挑战：提供收敛保证，以表征达到预定义容差所需的最坏情况迭代次数，并在固定迭代次数后上界一个性能指标。 这些保证对于计算时间有限的在线优化问题特别有用，因为在这些情况下现有的性能保证通常不可用或过于保守。 我们将收敛分析问题表述为基于有限采样参数实例集的场景优化程序。 利用场景优化理论中的工具，使我们能够推导出满足给定容差水平所需的迭代次数的概率保证。 利用场景优化的最新进展，我们进一步引入一种松弛方法，以在迭代次数和违反收敛标准阈值的风险之间进行权衡。 此外，我们通过将固定迭代优化问题转化为场景优化程序，分析了解决方案精度与时间效率之间的权衡。 数值模拟展示了我们方法在提供可靠的概率收敛保证以及评估解决方案精度与计算成本之间权衡方面的有效性。 显示英文摘要

摘要： We propose a data-driven method to establish probabilistic performance guarantees for parametric optimization problems solved via iterative algorithms. Our approach addresses two key challenges: providing convergence guarantees to characterize the worst-case number of iterations required to achieve a predefined tolerance, and upper bounding a performance metric after a fixed number of iterations. These guarantees are particularly useful for online optimization problems with limited computational time, where existing performance guarantees are often unavailable or unduly conservative. We formulate the convergence analysis problem as a scenario optimization program based on a finite set of sampled parameter instances. Leveraging tools from scenario optimization theory enables us to derive probabilistic guarantees on the number of iterations needed to meet a given tolerance level. Using recent advancements in scenario optimization, we further introduce a relaxation approach to trade the number of iterations against the risk of violating convergence criteria thresholds. Additionally, we analyze the trade-off between solution accuracy and time efficiency for fixed-iteration optimization problems by casting them into scenario optimization programs. Numerical simulations demonstrate the efficacy of our approach in providing reliable probabilistic convergence guarantees and evaluating the trade-off between solution accuracy and computational cost.