  In this paper, we propose a novel approach to solving optimization problems
by reformulating the optimization problem into a dynamical system, followed by
the adaptive spectral Koopman (ASK) method. The Koopman operator, employed in
our approach, approximates the evolution of an ordinary differential equation
(ODE) using a finite number of eigenfunctions and eigenvalues. We begin by
providing a brief overview of the Koopman operator and the ASK method.
Subsequently, we adapt the ASK method for solving a general optimization
problem. Moreover, we provide an error bound to aid in understanding the
performance of the proposed approach, marking the initial step in a more
comprehensive numerical analysis. Experimentally, we demonstrate the
applicability and accuracy of our method across a diverse range of optimization
problems, including min-max problems. Our approach consistently yields smaller
gradient norms and higher success rates in finding critical points compared to
state-of-the-art gradient-based methods. We also observe the proposed method
works particularly well when the dynamical properties of the system can be
effectively modeled by the system's behaviors in a neighborhood of critical
points.
