Fast Convergence of Inertial Multiobjective Gradient-like Systems with Asymptotic Vanishing Damping

TL;DR

This system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping and it is proved that its bounded solutions converge weakly to weakly Pareto optimal points.

Abstract

We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas laid out in [H. Attouch and G. Garrigos, Multiobjective optimization: an inertial approach to Pareto optima, preprint, arXiv:1506.02823, 201]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order $O(t^{-2})$ for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization.

Figures (2)

• Figure 1: Trajectories $x$ and inequalities $u_0(x(t)) \le \frac{t_0^2 u_0(x_0) + 2(\alpha - 1)R}{t^2}$ for different values of $\alpha \in \{3, 10, 50, 100 \}$.
• Figure 2: Trajectories $x$ and inequalities $u_0(x(t)) \le \frac{t_0^2 u_0(x_0) + 2(\alpha - 1)R}{t^2}$ for different values of $\alpha \in \{3, 10, 50, 100 \}$.

Theorems & Definitions (64)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Remark 2.5
• Example 2.6
• Lemma 2.7
• proof
• Proposition 3.1
• proof