WIAS Preprint No. 2230, (2016)

(Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution


  • van Ackooij, Wim
  • Henrion, René

2010 Mathematics Subject Classification

  • 90C15


  • stochastic optimization, gradients of probability functions, spheric radial decomposition, multivariate Gaussian distribution, Clarke subdifferential, Mordukhovich subdifferential, probabilistic constraint


We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven.

Appeared in

  • SIAM ASA J. Uncertain. Quantif., 5 (2017) pp. 63--87.

Download Documents