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

Monotonicity for Multiobjective Accelerated Proximal Gradient Methods