preprint
Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems
Konstantin
Sonntag
author 56399
Sebastian
Peitz
author 474270000-0002-3389-793X
101
department
655
department
We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like
dynamical system in the multiobjective setting, whose trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent by two to three orders of magnitude.
2022
eng
arXiv:2207.12707
2207.12707
Sonntag, Konstantin, and Sebastian Peitz. “Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems.” <i>ArXiv:2207.12707</i>, 2022.
K. Sonntag and S. Peitz, “Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems,” <i>arXiv:2207.12707</i>. 2022.
K. Sonntag, S. Peitz, ArXiv:2207.12707 (2022).
Sonntag, Konstantin, and Sebastian Peitz. “Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems.” <i>ArXiv:2207.12707</i>, 2022.
Sonntag K, Peitz S. Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems. <i>arXiv:220712707</i>. Published online 2022.
@article{Sonntag_Peitz_2022, title={Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems}, journal={arXiv:2207.12707}, author={Sonntag, Konstantin and Peitz, Sebastian}, year={2022} }
Sonntag, K., & Peitz, S. (2022). Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems. In <i>arXiv:2207.12707</i>.
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