  We consider quantile optimization of black-box functions that are estimated
with noise. We propose two new iterative three-timescale local search
algorithms. The first algorithm uses an appropriately modified
finite-difference-based gradient estimator that requires $2d$ + 1 samples of
the black-box function per iteration of the algorithm, where $d$ is the number
of decision variables (dimension of the input vector). For higher-dimensional
problems, this algorithm may not be practical if the black-box function
estimates are expensive. The second algorithm employs a
simultaneous-perturbation-based gradient estimator that uses only three samples
for each iteration regardless of problem dimension. Under appropriate
conditions, we show the almost sure convergence of both algorithms. In
addition, for the class of strongly convex functions, we further establish
their (finite-time) convergence rate through a novel fixed-point argument.
Simulation experiments indicate that the algorithms work well on a variety of
test problems and compare well with recently proposed alternative methods.
