Application of the Nosé-Hoover method to optimization problems

Area of Science:

• Computational Mathematics
• Applied Physics

Background:

• Continuous optimization problems often face a trade-off between solution accuracy and search speed.
• Existing methods struggle to simultaneously satisfy the need for high-probability candidate identification and rapid feasible region exploration.

Purpose of the Study:

• To propose a novel method for continuous optimization problems that reconciles the competing requirements of solution accuracy and search efficiency.
• To adapt the Nosé-Hoover dynamics for optimization by redefining physical system coordinates and potential functions.

Main Methods:

• Utilized the Nosé-Hoover equation, treating physical system coordinates as optimization decision variables.
• Replaced the traditional potential function with a term involving temperature and the logarithm of a density function.
• Engineered the density function to guide orbital trajectories towards low objective function values and independently control orbital speed.

Main Results:

• Demonstrated that the Nosé-Hoover dynamics, under ergodicity assumptions, can achieve desired visiting weights and orbital speeds through long-time limits.
• Numerical simulations confirmed the finite-time validity of the theoretical properties.
• Validated the applicability of the proposed Nosé-Hoover optimization method for practical problems.

Conclusions:

• The proposed Nosé-Hoover method effectively balances the probability of finding optimal solutions with the speed of searching within feasible regions.
• The method offers a compatible solution for continuous optimization, addressing limitations of prior approaches.
• Confirmed practical applicability and finite-time performance through simulations.