  Quantile is a popular performance measure for a stochastic system to evaluate
its variability and risk. To reduce the risk, selecting the actions that
minimize the tail quantiles of some loss distributions is typically of interest
for decision makers. When the loss distribution is observed via simulations,
evaluating and optimizing its quantile functions can be challenging, especially
when the simulations are expensive, as it may cost a large number of simulation
runs to obtain accurate quantile estimators. In this work, we propose a
multi-level metamodel (co-kriging) based algorithm to optimize quantile
functions more efficiently. Utilizing non-decreasing properties of quantile
functions, we first search on cheaper and informative lower quantiles which are
more accurate and easier to optimize. The quantile level iteratively increases
to the objective level while the search has a focus on the possible promising
regions identified by the previous levels. This enables us to leverage the
accurate information from the lower quantiles to find the optimums faster and
improve algorithm efficiency.
