Quantum Dynamics of Optimization Problems
This work aims to provide a new theoretical basis for studying optimization algorithms using dynamic methods, potentially benefiting researchers in algorithm design and analysis.
This paper transforms optimization problems into constrained state quantum problems by establishing a Schrödinger equation where the objective function acts as potential energy. It then uses this quantum interpretation to derive a Fokker-Planck equation for the time evolution of optimization algorithms, providing a basic iterative structure for these algorithms.
In this letter, by establishing the Schrödinger equation of the optimization problem, the optimization problem is transformed into a constrained state quantum problem with the objective function as the potential energy. The mathematical relationship between the objective function and the wave function is established, and the quantum interpretation of the optimization problem is realized. Under the black box model, the Schrödinger equation of the optimization problem is used to establish the kinetic equation, i.e., the Fokker-Planck equation of the time evolution of the optimization algorithm, and the basic iterative structure of the optimization algorithm is given according to the interpretation of the Fokker-Planck equation. The establishment of the Fokker-Planck equation allows optimization algorithms to be studied using dynamic methods and is expected to become an important theoretical basis for algorithm dynamics.