Fast Accelerated Proximal Gradient Method with New Extrapolation Term for Multiobjective Optimization
Chengzhi Huang
TL;DR
This work develops a fast accelerated proximal gradient framework for multiobjective optimization by introducing a novel extrapolation coefficient within a dedicated extrapolation term. For the smooth case, it proves a sublinear convergence rate $O\left(1/k^2\right)$ under mild initial assumptions on the Lipschitz constant, while extending to nonsmooth problems via a smoothing strategy that yields an $O\left(1/k\right)$ rate. A smoothing accelerated proximal gradient method with extrapolation (SAPGM) and its convergence analysis for both smooth and nonsmooth MOPs are presented, including detailed update rules that avoid line searches. Theoretical results are complemented by numerical experiments demonstrating effective Pareto-front approximation and practical performance on tri-objective problems. Overall, the paper provides a robust accelerated framework for MOPs with provable convergence guarantees under mild conditions and practical smoothing techniques for nonsmooth cases.
Abstract
In this paper, we propose a novel extrapolation coefficient scheme within a new extrapolation term and develop an accelerated proximal gradient algorithm. We establish that the algorithm achieves a sublinear convergence rate. The proposed scheme only requires the Lipschitz constant estimate sequence to satisfy mild initial conditions, under which a key equality property can be derived to support the convergence analysis. Numerical experiments are provided to demonstrate the effectiveness and practical performance of the proposed method.