  When minimizing a multiobjective optimization problem (MOP) using
multiobjective gradient descent methods, the imbalances among objective
functions often decelerate the convergence. In response to this challenge, we
propose two types of the Barzilai-Borwein proximal gradient method for
multi-objective composite optimization problems (BBPGMO). We establish
convergence rates for BBPGMO, demonstrating that it achieves rates of
$O(\frac{1}{\sqrt{k}})$, $O(\frac{1}{k})$, and $O(r^{k})(0<r<1)$ for
non-convex, convex, and strongly convex problems, respectively. Furthermore, we
show that BBPGMO exhibits linear convergence for MOPs with several linear
objective functions. Interestingly, the linear convergence rate of BBPGMO
surpasses the existing convergence rates of first-order methods for MOPs, which
indicates its enhanced performance and its ability to effectively address
imbalances from theoretical perspective. Finally, we provide numerical examples
to illustrate the efficiency of the proposed method and verify the theoretical
results.
