Exploiting Low-Dimensional Structure in Optimization Under Uncertainty
In computational science, optimization under uncertainty (OUU) provides a new methodology for building designs reliable under a variety of conditions with improved efficiency over a traditional, safety factor based approach. However, the resulting optimization problems can be challenging. For example, chance constraints bounding the probability that an arbitrary function exceeds a threshold are difficult; in absence of exploitable structure, these require estimation via Monte Carlo which is both noisy and expensive. In this talk I present joint work with Paul Constantine on solving OUU problems with chance constraints using a response surface approach with application to multi-physics model of a jet nozzle.
Our approach requires solving new optimization problems at several steps: building response surfaces from (1) polynomial ridge approximations with samples chosen using (2) a new sampling strategy and (3) exploiting structure to replace chance constraints with bounding linear inequality constraints. For (1), we show how a polynomial ridge approximation can be constructed by solving an optimization problem over the Grassmann manifold using a Gauss-Newton approach. For (2), we build a new sampling strategy based on a variable metric induced by the Lipschitz matrix: a new analog to the scalar Lipschitz constant that can reduce the effective intrinsic dimension. We show how the Lipschitz matrix can be estimated using either samples or gradients using a semidefinite program. For (3), we show how to exploit the low-dimensional structure in our response surfaces allows us to conservatively approximate the chance constraints with inequalities with a data-driven safety factor. This yields a linear program approximating the original OUU problem. This approach requires few expensive function evaluations to construct a design that, although not optimal, satisfies the chance-constraints with small probability of failure in our jet nozzle application.