Probabilistic learning for efficient optimization under risk constraints

International audienceIn this presentation, we tackle the challenge of mitigating the high cost of accurately estimating the expectations appearing in expressions of the objective and constraint functions. In order to facilitate the optimization task, we introduce two complementary perspectives. First, we consider jointly the decision variables and the quantities of interest (QoI) whose expectations form the constraints and objective functions. The second perspective recognizes that the mode input and outputs, as well as decision variables, are all related by the same physics (or black-box) constraints. The locus of these samples is a manifold. We then rely on diffusion manifold theory concepts to construct an algebraic basis for this manifold, following which a projected Ito equation is constructed that samples on this manifold without any further recourse to the (expensive) function evaluator. This Ito equation is constructed with an invariant measure specified by the distribution of the initial sample, thus controlling the probability measure on the manifold. With the sampling challenge thus addressed, optimization under uncertainty, with various types of probabilistic constraints, can be pursued without sacrificing accuracy or the form of of the constraints (eg replacing a probability requirement by a standard deviation requirement). Demonstrations on a number of mathematical and physics problems will be shown