We analyze an inner approximation scheme for probability maximization. The approach was proposed in Fabian, Csizmas, Drenyovszki, Van Ackooij, Vajnai, Kovacs, Szantai (2018) Probability maximization by inner approximation, Acta Polytechnica Hungarica 15:105-125, as an analogue of a classic dual approach in the handling of probabilistic constraints. Even a basic implementation of the maximization scheme proved usable and endured noise in gradient computations without any special effort. Moreover the speed of convergence was not affected by approximate computation of test points. This robustness was then explained in an idealized setting, considering a globally well-conditioned objective function. Here we work out convergence proofs for a logconcave distribution, specifically, for a normal distribution. The main result of the present paper is that the procedure gains traction as an optimal solution is approached.
Updated version: https://link.springer.com/article/10.1007/s10100-020-00697-3