A second-order dynamical approach with variable damping to nonconvex smooth minimization

Area of Science:

• Optimization Theory
• Dynamical Systems
• Non-convex Analysis

Background:

• Minimizing non-convex functions is challenging due to local minima.
• Second-order dynamical systems offer potential for efficient optimization.
• Nesterov's accelerated gradient method provides a benchmark for convex optimization.

Purpose of the Study:

• To investigate a second-order dynamical system with variable damping for non-convex function minimization.
• To analyze the convergence properties of this dynamical system.
• To establish convergence rates based on the Kurdyka-Lojasiewicz property.

Main Methods:

• Formulating a second-order dynamical system inspired by Nesterov's accelerated gradient method.
• Applying regularization to the objective function.
• Analyzing the system's trajectory convergence under the Kurdyka-Lojasiewicz property.

Main Results:

• The generated trajectory converges to a critical point under specific regularization conditions.
• Convergence rates are provided, dependent on the Lojasiewicz exponent.
• The variable damping mechanism is shown to be effective.

Conclusions:

• The proposed second-order dynamical system is a viable approach for non-convex function minimization.
• The Kurdyka-Lojasiewicz property is crucial for guaranteeing convergence.
• The study offers theoretical insights into the convergence behavior of accelerated methods for non-convex problems.