Probability estimation via policy restrictions, convexification, and approximate sampling

This paper develops various optimization techniques to estimate probability of events where the optimal value of a convex program, satisfying certain structural assumptions, exceeds a given threshold. First, we relate the search of affine/polynomial policies for the robust counterpart to existing relaxation hierarchies in MINLP (Lasserre in Proceedings of the international congress of mathematicians (ICM 2018), 2019; Sherali and Adams in A reformulation-linearization technique for solving discrete and continuous nonconvex problems, Springer, Berlin). Second, we leverage recent advances in Dworkin et al. (in: Kaski, Corander (eds) Proceedings of the seventeenth international conference on artificial intelligence and statistics, Proceedings of machine learning research, PMLR, Reykjavik, 2014), Gawrychowski et al. (in: ICALP, LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum fur Informatik, 2018) and Rizzi and Tomescu (Inf Comput 267:135-144, 2019) to develop techniques to approximately compute the probability binary random variables from Bernoulli distributions belong to a specially-structured union of sets. Third, we use convexification, robust counterpart, and chance-constrained optimization techniques to cover the event set of interest with such set unions. Fourth, we apply our techniques to the network reliability problem, which quantifies the probability of failure scenarios that cause network utilization to exceed one. Finally, we provide preliminary computational evaluation of our techniques on test instances for network reliability.