Efficient treatment of uncertainty in numerical optimization

We have combined the methods of probabilistic risk analysis and optimization to devise a technique suitable for the efficient treatment of uncertainties (or the effects of random fluctuations) in the design and analysis of mathematically describable processes. The key step is the approximation, by a multivariable Taylor series expansion, of the influence of random variables on the objective function. Statistical averaging of this expansion leads to a description of the objective function in terms of the moments of the random variables. Knowledge or estimation of these moments allows the optimization to be carried out using standard calculus based techniques. An example is treated with three variations to illustrate the use of this technique for nonlinear sets of equations and objective functions. The method presented here is applicable to process models in manufacturing, systems analysis, and risk analysis.