  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.
