  We investigate an empirical quantile estimation approach to solve
chance-constrained nonlinear optimization problems. Our approach is based on
the reformulation of the chance constraint as an equivalent quantile constraint
to provide stronger signals on the gradient. In this approach, the value of the
quantile function is estimated empirically from samples drawn from the random
parameters, and the gradient of the quantile function is estimated via a
finite-difference approximation on top of the quantile-function-value
estimation. We establish a convergence theory of this approach within the
framework of an augmented Lagrangian method for solving general nonlinear
constrained optimization problems. The foundation of the convergence analysis
is a concentration property of the empirical quantile process, and the analysis
is divided based on whether or not the quantile function is differentiable. In
contrast to the sampling-and-smoothing approach used in the literature, the
method developed in this paper does not involve any smoothing function and
hence the quantile-function gradient approximation is easier to implement and
there are less accuracy-control parameters to tune. We demonstrate the
effectiveness of this approach and compare it with a smoothing method for the
quantile-gradient estimation. Numerical investigation shows that the two
approaches are competitive for certain problem instances.
