Gradient formulae for probability functions depending on a heterogenous family of constraints - van Ackooij, Wim; Pérez-Aros, Pedro (Open Journal of Mathematical Optimization)
Probability functions measure the degree of satisfaction of certain constraints that are impacted by decisions and uncertainty. Such functions appear in probability or chance constraints ensuring that the degree of satisfaction is sufficiently high. These constraints have become a very popular modelling tool and are indeed intuitively easy to understand. Optimization problems involving probabilistic constraints have thus arisen in many sectors of the industry, such as in the energy sector. Finding an efficient solution methodology is important and first order information of probability functions play a key role therein. In this work we are motivated by probability functions measuring the degree of satisfaction of a potentially heterogenous family of constraints. We suggest a framework wherein each individual such constraint can be analyzed structurally. Our framework then allows us to establish formulae for the generalized subdifferential of the probability function itself. In particular we formally establish a (sub)-gradient formulæ for probability functions depending on a family of non-convex quadratic inequalities. The latter situation is relevant for gas-network applications.