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Multiobjective Accelerated Gradient-like Flow with Asymptotic Vanishing Normalized Gradient

Yingdong Yin

TL;DR

This work extends accelerated gradient dynamics to multiobjective optimization by introducing a gradient-like flow with asymptotically vanishing normalized gradients (MAVNG). Using Lyapunov analysis and differential inclusion techniques, it establishes convergence rates of $O(1/t^2)$ (and $O(\ln^2 t / t^2)$ under a marginally weaker condition) for trajectory solutions, and shows convergence to weak Pareto optima under convexity. A discretization yields the Multiobjective Fast Inertial Search Direction Correction (MFISC) method with a convergence rate of $O(\ln^2 k / k^2)$ in terms of a merit function, and all limit points are weakly Pareto optimal. Numerical experiments demonstrate faster convergence and favorable Pareto-front approximation compared with existing dynamical systems and methods, validating the theoretical results and highlighting practical efficiency considerations. The framework offers a principled path to accelerate multiobjective optimization while preserving Pareto convergence guarantees, with potential extensions to balanced gradient flows and time-scaling strategies.

Abstract

This paper generalizes the dynamical system proposed by Wang et al. [Siam. J. Sci. Comput., 2021] to multiobjective optimization by investigating a multiobjective accelerated gradient-like flow with asymptotically vanishing normalized gradient. Using Lyapunov analysis, we obtain convergence rates of $O(1/t^2)$ and $O(\ln^2 t / t^2)$ for the trajectory solution under two distinct parameter selections. Under certain assumptions, we further prove that the trajectory solution of this gradient flow converges to a weak Pareto solution for convex multiobjective optimization problems. Through corresponding discretization, we derive a new class of multiobjective gradient methods achieving a convergence rate of $O(\ln^2 k / k^2)$. Additionally, numerical experiments validate the theoretical results, demonstrating that this gradient flow outperforms other existing dynamical systems in the literature regarding convergence speed, and our algorithm exhibits corresponding advantages.

Multiobjective Accelerated Gradient-like Flow with Asymptotic Vanishing Normalized Gradient

TL;DR

This work extends accelerated gradient dynamics to multiobjective optimization by introducing a gradient-like flow with asymptotically vanishing normalized gradients (MAVNG). Using Lyapunov analysis and differential inclusion techniques, it establishes convergence rates of (and under a marginally weaker condition) for trajectory solutions, and shows convergence to weak Pareto optima under convexity. A discretization yields the Multiobjective Fast Inertial Search Direction Correction (MFISC) method with a convergence rate of in terms of a merit function, and all limit points are weakly Pareto optimal. Numerical experiments demonstrate faster convergence and favorable Pareto-front approximation compared with existing dynamical systems and methods, validating the theoretical results and highlighting practical efficiency considerations. The framework offers a principled path to accelerate multiobjective optimization while preserving Pareto convergence guarantees, with potential extensions to balanced gradient flows and time-scaling strategies.

Abstract

This paper generalizes the dynamical system proposed by Wang et al. [Siam. J. Sci. Comput., 2021] to multiobjective optimization by investigating a multiobjective accelerated gradient-like flow with asymptotically vanishing normalized gradient. Using Lyapunov analysis, we obtain convergence rates of and for the trajectory solution under two distinct parameter selections. Under certain assumptions, we further prove that the trajectory solution of this gradient flow converges to a weak Pareto solution for convex multiobjective optimization problems. Through corresponding discretization, we derive a new class of multiobjective gradient methods achieving a convergence rate of . Additionally, numerical experiments validate the theoretical results, demonstrating that this gradient flow outperforms other existing dynamical systems in the literature regarding convergence speed, and our algorithm exhibits corresponding advantages.

Paper Structure

This paper contains 30 sections, 31 theorems, 133 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Accelerated gradient methods and flows for multiobjective optimization
  3. Fast inertial seaech direction correction algorithm
  4. Motivation
  5. Contributions
  6. Organization
  7. Preliminary
  8. Notation
  9. Pareto optimality and necessary condition
  10. Merit function
  11. Assumption
  12. Existence of solutions
  13. Existence of solutions to the differential inclusion \ref{['eq:DI']}
  14. Existence of solutions to the differential equation \ref{['eq:Cauchy-Problem']}
  15. Asymptotic analysis for gradient-like flow
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $\varphi:\mathbb R^n \to \,{\mathbb R}$ be defined as in eq:meritfunction with lower semicontinous functions $f_i$ for all $i=1,\cdots,m$. Then for eq:MOP, we have

Figures (6)

  • Figure 1: FISC-nes iteration diagram
  • Figure 2: MFISC iteration diagram
  • Figure 3: For the quadratic programming problems, the trajectories and changes in function values of $\rm MAVNG$ and $\rm MAVD$. The red line corresponds to MAVNG, and the blue line corresponds to MAVD.
  • Figure 4: For the non-quadratic programming problems, the trajectories and changes in function values of $\rm MAVNG$ and $\rm MAVD$. The red line corresponds to MAVNG, and the blue line corresponds to MAVD.
  • Figure 5: The image sets generated by the two comparative algorithms for the convex problems JOS1a, SDa and TOI4a. (a)--(c) correspond to MFISC; (d)--(f) correspond to AccG.
  • ...and 1 more figures

Theorems & Definitions (69)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • remark thmcounterremark
  • Theorem 2.1
  • proof
  • definition thmcounterdefinition
  • remark thmcounterremark
  • Theorem 2.2
  • proof
  • Corollary 2.1
  • ...and 59 more