An accelerated proximal gradient method for multiobjective optimization

TL;DR

This paper introduces an accelerated proximal gradient method for multiobjective optimization, extending FISTA to handle multiple objectives without scalarization, and proves its convergence rate with practical numerical validation.

Contribution

It develops a multiobjective generalization of FISTA with a novel subproblem solution, providing the first accelerated method with proven convergence rates for this setting.

Findings

Achieves a global convergence rate of O(1/k^2)

Provides an efficient dual-based subproblem solver

Validates the method through numerical experiments

Abstract

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.