Talks and presentations

First-Order Methods in Convex Optimization: From Discrete to Continuous and Vice-versa

September 27, 2024

Invited Talk, The 8-th Chinese–German Workshop on Computational and Applied Mathematics, School of Mathematics, Sichuan University, Chengdu, P.R.China

Recently, continuous-time approach has been widely used to study first-order methods (FOMs) for structured convex optimization problems, such as Nesterov accelerated gradient (NAG), alternating method of multiplies (ADMM) and primal-dual hybrid gradient (PDHG) From the continuous point of view, the discrete iterative sequences actually correspond to the trajectories of some ordinary differential equations (ODEs), and the use of Inertia/Momentum and Gradient Correction usually leads to second-order information in time and space. In this talk, we shall present a unified ODE2OPT framework, which gives a systematic way for deriving the continuous-time ODE of a given FOM and provides also an alternative way for designing and analyzing FOMs by using the tool of Lyapunov functional and numerical analysis.

Effcient primal-dual algorithms for optimal transport-like problems

July 06, 2022

Postdoc Research Report, School of Mathematical Sciences, Peking University, Beijing, P.R.China

In this report, we focus on effcient algorithms for solving a large class of optimal transport-like problems. We will first introduce some primal-dual flow models and equip them with proper Lyapunov functions that possess exponential decay property. Then, based on time discretizations of the continuous dynamics, we obtain new primal-dual methods and prove the nonergodic ((super-)linear or sublinear)convergence rates. Besides, by exploring the special structure, the inner problems of the proposed methods either admit closed solution or can be solved by the semi-smooth Newton iteration with algebraic multigrid method. Moreover, numerical experiments are provided to validate the effciency of our algorithms.

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