SGER: Characterizing Sampling Error for Optimization Under Uncertainty - A Fractal Geometry Approach

这项探索性研究小额资助（SGER）的目标是开发创新和强大的方法来解决不确定性下的决策问题，并应用于大规模的设计、制造、规划和管理相关问题。然而，在存在不确定性的情况下对性能进行有效评估是关键的第一步。不确定条件下的决策问题是一个随机优化问题，其本质是对由不确定的输入参数分布采样得到的输入参数集，由多个确定性模拟构造的一个或多个概率输出函数进行约束优化。该方法的强大和有用性已被主要研究人员证明，但该方法的计算负担可能是极端的，并且取决于用于表征参数不确定性的样本量。采样精度对提高这些优化算法的效率起着重要的作用。提出的方法旨在填补在评估采样误差带宽宽度方面的一个重要空白。它利用分形几何的概念来推导误差估计。在参数不确定性云中的鲁棒决策在许多应用中是至关重要的。这些不确定性可能来自操作、环境和市场参数（例如，环境温度、客户需求）、错误的模型表示（模型简化和假设）以及模型参数和数据（例如，反应常数、物理性质、测量误差）。此外，决策者必须处理离散决策，例如是否选择特定选项，以及在连续空间中的决策，例如工厂操作温度的选择。

The objectives of this Small Grant for Exploratory Research (SGER) are to develop innovative and powerful approaches for solving problems of decision making under uncertainty applied to large scale design, manufacturing, planning, and management related problems. However, efficient estimation of performance in the presence of uncertainty is a critical first step. The problem of decision making under uncertainty is posed as a stochastic optimization problem, which fundamentally involves constrained optimization of one or more probabilistic output functions constructed from multiple deterministic simulations for input parameter sets obtained by sampling uncertain input parameter distributions. The power and usefulness of the approach has been demonstrated by the Principal Investigators, but the computational burden of this approach can be extreme and depends on the sample size used for characterizing the parametric uncertainties. Sampling accuracy plays an important role in enhancing the efficiency of these optimization algorithms. The proposed approach is aimed at filling an important void in assessing the width of the sampling error-bandwidth in specialized Monte Carlo approaches. It draws on concepts from fractal geometry to derive error estimates.Robust decision making amidst a cloud of parameter uncertainties is of fundamental importance in many applications. These uncertainties can arise from operational, environmental, and market parameters (e.g., ambient temperature, customer demand), erroneous model representations (model simplifications and assumptions),and model parameters and data (e.g. reaction constants, physical properties, errors in measurements), to name a few. Furthermore, a decision maker has to deal with discrete decisions, like whether to choose a particular option or not, and also with decisions on a continuous space such as the choice of the operating temperature in a plant.