Time Rescaling of a Primal-Dual Dynamical System with Asymptotically Vanishing Damping

Area of Science:

• Optimization Theory
• Convex Analysis
• Dynamical Systems

Background:

• Minimizing convex functions under linear constraints is a fundamental problem in optimization.
• Existing second-order dynamical systems offer convergence but may lack tunable parameters for rate improvement.

Purpose of the Study:

• To develop and analyze a modified second-order dynamical system for convex minimization with linear equality constraints.
• To investigate the impact of a time rescaling parameter on convergence rates.
• To establish theoretical convergence guarantees for primal-dual trajectories.

Main Methods:

• Formulation of a second-order dynamical system with an asymptotically vanishing damping term.
• Analysis of the system's trajectories using convergence rate analysis.
• Investigation of weak convergence properties for primal-dual solutions.
• Numerical experiments to compare with existing dynamical systems.

Main Results:

• Demonstrated fast convergence for primal-dual gap, feasibility measure, and objective function value.
• Established that convergence rates are dependent on the time rescaling parameter, allowing for potential improvement.
• Proved weak asymptotic convergence of primal-dual trajectories to optimal solutions under Lipschitz gradient conditions.
• Showcased improved convergence rates for primal gradients and dual adjoint operators.

Conclusions:

• The proposed time-rescaled second-order dynamical system provides an effective and improvable method for convex optimization.
• The ability to tune convergence rates via the rescaling parameter offers practical advantages.
• Numerical results validate the theoretical findings and highlight the system's performance compared to alternatives.