Host: Jin-Peng Liu 刘锦鹏

Abstract:

This talk explores some recent advancements in first-order optimization algorithms for composite optimization and is divided into two parts: convergence analysis of FISTA and practical application of an ADMM-like algorithm in data assimilation.

In the first part, we refine the convergence analysis of FISTA for convex composite optimization. By introducing a tighter pivotal inequality, we improve the convergence rate estimates for subgradient norm minimization. Furthermore, we demonstrate that FISTA achieves linear convergence when applied to strongly convex functions, even without prior knowledge of the strong convexity modulus. This analysis is based on a Lyapunov function framework and high-resolution differential equations (ODEs) that characterize the acceleration phenomenon.

The second part introduces a Parallel Linearized Proximal ADMM algorithm designed for 4D Variational Data Assimilation (4D-Var). While traditional gradient-based methods for 4D-Var are often sensitive to initial guesses and difficult to parallelize, our proposed ADMM approach allows for parallel-in-time implementation. We provide numerical evidence of the algorithm's efficacy using complex nonlinear systems, including the Lorenz system, Burgers’ equation, and 2D vorticity equations. Finally, we discuss new convergence results for proximal multiblock ADMM in the context of nonconvex dynamics-constrained optimization.

Bio:

Bowen Li is a final-year PhD student at the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences. His research focuses on the development and theoretical analysis of first-order algorithms for composite optimization and their applications in large-scale scientific computing.

His contributions have been published in journals including the Journal of Computational Physics, SIAM Journal on Optimization, Applied and Computational Harmonic Analysis, and the Journal of the Operations Research Society of China.