Accelerated Proximal Gradient Method with Backtracking for Multiobjective Optimization

TL;DR

This paper introduces a novel backtracking strategy based on FISTA for multiobjective optimization, enabling convergence guarantees without prior knowledge of the Lipschitz constant, and achieves an $O(1/k^2)$ rate.

Contribution

It presents a new backtracking approach for accelerated proximal gradient methods tailored to multiobjective problems, addressing unknown Lipschitz constants and ensuring convergence.

Findings

Achieves $O(1/k^2)$ convergence rate.

Effectively estimates parameters without non-increasing constraints.

Provides theoretical guarantees under mild assumptions.

Abstract

This paper proposes a new backtracking strategy based on the FISTA accelerated algorithm for multiobjective optimization problems. The strategy focuses on solving the problem of Lipschitz constant being unknown. It allows estimate parameter updates non-increasingly. Furthermore, the proposed strategy effectively avoids the limitation in convergence proofs arising from the non-negativity of the auxiliary sequence, thus providing a theoretical guarantee for its performance. We demonstrate that, under relatively mild assumptions, the algorithm achieves the convergence rate of $O(1/k2)$.