This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ($O(1 / k^2)$), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

翻译：本文提出了一种用于多目标优化的加速近端梯度方法，其中每个目标函数由连续可微凸函数与闭真凸函数之和构成。将无标量化处理的多目标一阶方法进行推广已被广泛研究，但提供具有精确收敛速度证明的加速方法仍是一个开放性问题。所提方法是标量优化中加速近端梯度方法（即快速迭代收缩阈值算法，FISTA）的多目标推广。该扩展成功的关键在于求解一个多目标情形特有的子问题。通过采用绩效函数衡量复杂度，我们得以证明所提方法的全局收敛速率（$O(1 / k^2)$）。此外，我们提出了一种通过子问题对偶表示进行高效求解的方式，并通过数值实验验证了所提方法的有效性。