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

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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.