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

Author

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  • Konstantin Sonntag

    (Paderborn University)

  • Sebastian Peitz

    (Paderborn University)

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, which 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.

Suggested Citation

  • Konstantin Sonntag & Sebastian Peitz, 2024. "Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 539-582, May.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:2:d:10.1007_s10957-024-02389-3
    DOI: 10.1007/s10957-024-02389-3
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    References listed on IDEAS

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    4. Mustapha El Moudden & Abdelkrim El Mouatasim, 2021. "Accelerated Diagonal Steepest Descent Method for Unconstrained Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 220-242, January.
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