Gradient estimates for gaussian distribution functions: Application to probabilistically constrained optimization problems

Abstract

We provide lower estimates for the norm of gradients of Gaussian distribution functions and apply the results obtained to a special class of probabilistically constrained optimization problems. In particular, it is shown how the precision of computing gradients in such problems can be controlled by the precision of function values for Gaussian distribution functions. Moreover, a sensitivity result for optimal values with respect to perturbations of the underlying random vector is derived. It is shown that the so-called maximal increasing slope of the optimal value with respect to the Kolmogorov distance between original and perturbed distribution can be estimated explicitly from the input data of the problem.