Energy-Stable Swarm-Based Inertial Algorithms for Optimization

Abstract We formulate the swarming optimization problem as a weakly coupled, dissipative dynamical system governed by a controlled energy dissipation rate and initial velocities that adhere to the nonequilibrium Onsager principle. In this formulation, agents’ inertia, positions, and masses are dynamically coupled. To enable rapid convergence toward equilibrium, we develop a class of efficient, energy-stable numerical algorithms for the dynamical system that preserve or enhance energy dissipation at the discrete level. At the equilibrium, the system reaches one of the lowest local minima explored by the agents, thereby improving the likelihood of identifying the global minimum. Numerical experiments confirm the effectiveness of the proposed approach, demonstrating significant advantages over traditional swarm-based gradient descent methods, especially when operating with a limited number of agents.