Augmented Lagrangian method for probabilistic optimization

We analyze nonlinear stochastic optimization problems with probabilistic constraints described by continuously differentiable non-convex functions. We describe the tangent and the normal cone to the level sets of the underlying probability function and provide new insight into their structure. Furthermore, we formulate fist order and second order conditions of optimality for these problems based on the notion of p-efficient points. We develop an augmented Lagrangian method for the case of discrete distribution functions. The method is based on progressive inner approximation of the level set of the probability function by generation of p-efficient points. Numerical experience is provided.