Adaptive probabilistic branch and bound with confidence intervals for level set approximation

We present a simulation optimization algorithm called probabilistic branch and bound with confidence intervals (PBnB with CI), which is designed to approximate a level set of solutions for a user-defined quantile. PBnB with CI is developed for both deterministic and noisy problems with mixed continuous and discrete variables. The quality of the results is statistically analyzed with order statistic techniques and confidence intervals are derived. Also, the number of samples and replications are designed to achieve a certain quality of solutions. When the algorithm terminates, it provides an estimation of the desired quantile with confidence intervals, and an approximation level set, including a statistically guaranteed set in the true desirable level set, a statistically pruned set, and a set which is not statistically specified. We also present numerical experiments with benchmark functions to visualize the algorithm and its capability.