Monotonicity for Multiobjective Accelerated Proximal Gradient Methods

TL;DR

This paper introduces a monotonic variant of the multiobjective FISTA algorithm, ensuring convergence and numerical stability, with proven $O(1/k^2)$ rate and practical benefits over existing methods.

Contribution

It proposes a new monotonic multiobjective FISTA variant that guarantees convergence and improves numerical stability compared to previous approaches.

Findings

Proven global convergence with rate $O(1/k^2)$.

Numerical experiments show advantages of monotonicity.

Extension of MFISTA to multiobjective problems.

Abstract

Accelerated proximal gradient methods, which are also called fast iterative shrinkage-thresholding algorithms (FISTA) are known to be efficient for many applications. Recently, Tanabe et al. proposed an extension of FISTA for multiobjective optimization problems. However, similarly to the single-objective minimization case, the objective functions values may increase in some iterations, and inexact computations of subproblems can also lead to divergence. Motivated by this, here we propose a variant of the FISTA for multiobjective optimization, that imposes some monotonicity of the objective functions values. In the single-objective case, we retrieve the so-called MFISTA, proposed by Beck and Teboulle. We also prove that our method has global convergence with rate $O(1/k^2)$, where $k$ is the number of iterations, and show some numerical advantages in requiring monotonicity.