Probabilistic learning on manifold for optimization under uncertainties

Plenary LectureInternational audienceThis paper presents a challenging problem devoted to the probabilistic learning on manifold for the optimization under uncertainties and a novel idea for solving it. The methodology belongs to the class of the statistical learning methods and allows for solving the probabilistic nonconvex constrained optimization with a fixed number of expensive function evaluations. It is assumed that the expensive function evaluator generates samples (defining a given dataset) that randomly fluctuate around a "manifold". The objective is to develop an algorithm that uses a number of expensive function evaluations at a level essentially equal to that of the deterministic problem. The methodology proposed consists in using an algorithm to generate additional samples in the neighborhood of this manifold from the joint probability distribution of the design parameters and of the random quantities that defined the objective and the constraint functions. This is achieved by using the probabilistic learning on manifold from the given dataset generated by the optimizer without performing additional expensive function evaluations. A statistical smoothing technique is developed for estimating the mathematical expectations in the computation of the objective and constraint functions at any point of the admissible set by using only the additional samples. Several numerical illustrations are presented for validating the proposed approach