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We investigate an empirical quantile estimation approach to solvechance-constrained nonlinear optimization problems. Our approach is based onthe reformulation of the chance constraint as an equivalent quantile constraintto provide stronger signals on the gradient. In this approach, the value of thequantile function is estimated empirically from samples drawn from the randomparameters, and the gradient of the quantile function is estimated via afinite-difference approximation on top of the quantile-function-valueestimation. We establish a convergence theory of this approach within theframework of an augmented Lagrangian method for solving general nonlinearconstrained optimization problems. The foundation of the convergence analysisis a concentration property of the empirical quantile process, and the analysisis divided based on whether or not the quantile function is differentiable. Incontrast to the sampling-and-smoothing approach used in the literature, themethod developed in this paper does not involve any smoothing function andhence the quantile-function gradient approximation is easier to implement andthere are less accuracy-control parameters to tune. We demonstrate theeffectiveness of this approach and compare it with a smoothing method for thequantile-gradient estimation. Numerical investigation shows that the twoapproaches are competitive for certain problem instances.

An Empirical Quantile Estimation Approach to Nonlinear Optimization Problems with Chance Constraints