Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems

TL;DR

This paper introduces accelerated inertial gradient algorithms for multiobjective convex optimization that improve convergence rates and efficiency, avoiding complex subproblem solutions and demonstrating significant speedups in tests.

Contribution

It develops novel inertial gradient-like dynamical systems and their discretizations for multiobjective optimization, incorporating Nesterov acceleration and subproblem avoidance.

Findings

Algorithms achieve weak Pareto optimality convergence.

Accelerated methods are 100-1000 times faster than standard steepest descent.

Avoiding subproblem solutions enhances computational efficiency.

Abstract

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.