CAM Colloquium: Drew Kouri (Sandia National Laboratories) - A primal-dual algorithm for large-scale risk minimization

Description

Abstract: Many science and engineering applications necessitate the optimization of systems described by partial differential equations (PDEs) with uncertain inputs including noisy physical parameters, unknown boundary or initial conditions, and unverifiable modeling assumptions. One can formulate such problems as risk-averse optimization problems in Banach space, which upon discretization, become enormous risk-averse stochastic programs. For many popular risk models including the coherent risk measures, the resulting risk-averse objective function is not differentiable. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, I present a general primal-dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by epigraphical regularization of risk measures and is closely related to the classical method of multipliers. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. I prove convergence of the algorithm even when the subproblem solves are performed inexactly and conclude my presentation with multiple PDE-constrained examples that demonstrate the efficiency of this method.

Bio: Drew Kouri is a staff member in the Optimization and Uncertainty Quantification Department at Sandia National Laboratories. He received his BS and MS (2008) in mathematics from Case Western Research University, and his MA (2010) and PhD (2012) in computational and applied mathematics from Rice University. Before joining Sandia, he was the J. H. Wilkinson Fellow at Argonne National Laboratory. His research focuses on the analysis and numerical solution of PDE-constrained optimization and stochastic programming problems.