Application of probabilistic ordinal optimization concepts to a continuous-variable probabilistic optimization problem

A very general and robust approach to solving optimization problems involving probabilistic uncertainty is through the use of probabilistic ordinal optimization. At each step in the optimization problem, improvement is based only on a relative ranking of the probabilistic merits of local design alternatives, rather than on crisp quantification of the alternatives. Thus, we simply ask the question: "Is that alternative better or worse than this one?" to some level of statistical confidence we require, not: "how much better or worse is that alternative to this one?". We illustrate an elementary application of probabilistic ordinal concepts in a 2-D optimization problem. Two uncertain variables contribute to uncertainty in the response function. We use a simple coordinate pattern search nongradient-based optimizer to step toward the statistical optimum in the design space. We also discuss more sophisticated implementations, and some of the advantages and disadvantages versus nonordinal approaches for optimization under uncertainty