Global Optimization via Softmin Energy Minimization

Non-convex optimization in high dimensions requires balancing gradient information with global exploration, a tension that Langevin dynamics and other gradient-informed methods address only partially. We introduce a gradient-based swarm method driven by a Softmin energy interaction $J_\beta(\mathbf{x})$, a smooth approximation of the minimum over the ensemble that couples particles through their relative energies. Combining the resulting gradient flow with Brownian noise, we obtain a dynamics that retains the efficiency of gradient steps while inheriting the global-search behavior of swarm methods. We show that this interaction always partitions the ensemble into particles descending toward local minima and particles automatically repelled toward maxima, and that the same interaction caps the Freidlin--Wentzell quasipotential barrier separating adjacent basins at a level of order $1/\beta$, independently of the true barrier height, strictly below the barrier faced by an overdamped Langevin particle in the same landscape. Numerical experiments on double-well, quadruple-well, and Lennard-Jones cluster benchmarks confirm the analysis and show improved reliability over Simulated Annealing, Langevin dynamics, and variants of Particle Swarm Optimization and Consensus Based Optimization; a high-dimensional Ackley benchmark marks the limit of the approach, where Particle Swarm Optimization is decisively faster.