An accelerated primal-dual flow for linearly constrained multiobjective optimization

TL;DR

This paper introduces an accelerated primal-dual flow for linearly constrained multiobjective optimization, providing a continuous-time model with exponential convergence analysis and a practical numerical scheme with sublinear rates.

Contribution

It extends the accelerated primal-dual flow to multiobjective problems, introduces a new merit function for convergence analysis, and develops an efficient numerical scheme with proven convergence rates.

Findings

Exponential decay rate established for the continuous model.

Sublinear convergence rates proved for the discrete scheme.

Numerical experiments demonstrate the method's effectiveness.

Abstract

In this paper, we propose a continuous-time primal-dual approach for linearly constrained multiobjective optimization problems. A novel dynamical model, called accelerated multiobjective primal-dual flow, is presented with a second-order equation for the primal variable and a first-order equation for the dual variable. It can be viewed as an extension of the accelerated primal-dual flow by Luo [arXiv:2109.12604, 2021] for the single objective case. To facilitate the convergence rate analysis, we introduce a new merit function, which motivates the use of the feasibility violation and the objective gap to measure the weakly Pareto optimality. By using a proper Lyapunov function, we establish the exponential decay rate in the continuous level. After that, we consider an implicit-explicit scheme, which yields an accelerated multiobjective primal-dual method with a quadratic subproblem, and prove the sublinear rates of the feasibility violation and the objective gap, under the convex case and the strongly convex case, respectively. Numerical results are provided to demonstrate the performance of the proposed method.