Adaptive Randomized Sketching for Dynamic Nonsmooth Optimization

Dynamic optimization problems arise in many applications, including optimal flow control, full waveform inversion, and medical imaging, where they are plagued by significant computational challenges. For example, memory is often a limiting factor on the size of problems one can solve since the evaluation of derivatives requires the entire state trajectory. Additionally, many applications employ nonsmooth regularizers such as the $$L^1$$ -norm or the total variation as well as auxiliary constraints on the optimization variables. In this chapter, we introduce a novel trust-region algorithm for minimizing the sum of a smooth, nonconvex function and a nonsmooth, convex function that addresses these two challenges. Our algorithm employs randomized sketching to store a compressed version of the state trajectory for use in derivative computations. By allowing the trust-region algorithm to adaptively learn the rank of the state sketch, we arrive at a provably convergent method with near optimal memory requirements. We demonstrate the efficacy of our method on a parabolic PDE-constrained optimization problem with measure-valued control variables.