Mathematics Colloquium: An Approximation Scheme for Distributionally Robust Nonlinear Programming with Applications to PDE-Constrained Optimization under Uncertainty

Speaker:  Michael Ulbrich, Technical University Munich

Title:  An Approximation Scheme for Distributionally Robust Nonlinear Programming with Applications to PDE-Constrained Optimization under Uncertainty

Abstract:  We present a sampling-free approximation scheme for distributionally robust nonlinear optimization (DRO). The DRO problem can be written in a bilevel form that involves maximal (i.e., worst case) value functions of expectation of nonlinear functions that depend on the optimization variables and random parameters. The maximum values are taken over an ambiguity set of probability measures which is defined by moment constraints. To achieve a good compromise between tractability and accuracy we approximate nonlinear dependencies of the cost / constraint functions on the random parameters by quadratic Taylor expansions. This results in an approximate DRO problem which on the lower level then involves value functions of parametric trust-region problems and of parametric semidefinite programs. These value functions are in general nonsmooth with respect to the decision variables. Using trust-region duality, a barrier approach, and other techniques we construct gradient consistent smoothing functions for the value functions and show global convergence of a corresponding homotopy method. We discuss the application of our approach to PDE constrained optimization under uncertainty and present numerical results. This is joint work with Johannes Milz.

Date and Time:  March 31, 12:15 - 1:05PM

Registration is mandatory and deadline to register is March 30, 2022. Please register here