Barzilai-Borwein Proximal Gradient Methods for Multiobjective Composite Optimization Problems with Improved Linear Convergence

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.