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A primal-dual algorithm for large-scale risk minimization

Banff International Research Station for Mathematical Innovation and Discovery

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. The algorithm solves a sequence of smooth optimization problems using derivative-based methods and is provably convergent even when the subproblem solves are performed inexactly. I conclude my presentation with multiple PDE-constrained examples that demonstrate the efficiency of this method.

Mathematics; Calculus Of Variations And Optimal Control; Optimization; Partial Differential Equations; Control/optimization/operation Research

video/mp4

Author affiliation: Sandia National Laboratories

2021-08-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/79224

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/