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

这一探索性研究小额拨款(SGER)的目标是开发创新和强大的方法来解决不确定情况下的决策问题，适用于大规模设计、制造、规划和管理相关问题。然而，在存在不确定性的情况下有效地估计性能是关键的第一步。不确定条件下的决策问题本质上是一个随机优化问题，其实质是对输入参数集进行多次确定性模拟得到的一个或多个概率输出函数的约束最优化问题。该方法的能力和有效性已被首席研究者证明，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大小。采样精度对提高这些优化算法的效率起着重要作用。该方法旨在填补专门的蒙特卡罗方法在估计抽样误差带宽方面的一个重要空白。它利用分形几何中的概念来推导误差估计。在许多应用中，在参数不确定的云层中进行稳健的决策是非常重要的。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、测量误差等)。此外，决策者必须处理离散的决策，如是否选择特定的选项，以及关于连续空间的决策，如选择工厂的运行温度。

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.