Journal of the
Korean Mathematical Society JKMS

There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a $C^2$ strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.

Keywords: Shadowing property, sensitivity, asymptotic behavior, long time behavior, ADMM, optimization flow, optimization methods, convex programming, proximal method

MSC numbers: Primary 37N40; Secondary 90C31, 90C25

Supported by: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2016R1A5A1008055). The work of Y. M. Jung was supported by NRF of Korea (No. 2022R1A2C1010537). The work of B. Shin was supported by NRF of Korea (No. 2021R1C1C2005241). The work of S. Yun was supported by NRF of Korea (No. 2022R1A2C1011503).