  We introduce a novel method for non-convex optimization which is at the
interface between the swarm-based gradient-descent (SBGD) [J. Lu et. al.,
ArXiv:2211.17157; E.Tadmor and A. Zenginoglu, Acta Applicandae Math., 190,
2024] and Simulated Annealing (SA) [V. Cerny, J. optimization theory and appl.,
45:41-51, 1985; S.Kirkpatrick et. al., Science, 220(4598):671-680, 1983; S.
Geman and C.-R. Hwang, SIAM J. Control and Optimization, 24(5):1031-1043,
1986]. We follow the methodology of SBGD in which a swarm of agents, each
identified with a position, ${\mathbf x}$ and mass $m$, explores the ambient
space. The agents proceed in gradient descent direction, and are subject to
Brownian motion with annealing-rate dictated by a decreasing function of their
mass. Thus, instead of the SA protocol for time-decreasing temperature, we let
the swarm decide how to `cool down' agents, depending on their accumulated mass
over time. The dynamics of masses is coupled with the dynamics of positions:
agents at higher ground transfer (part of) their mass to those at lower ground.
Consequently, the swarm is dynamically divided between heavier, cooler agents
viewed as `leaders' and lighter, warmer agents viewed as `explorers'.
Mean-field convergence analysis and benchmark optimizations demonstrate the
effectiveness of the swarm-based method as a multi-dimensional global
optimizer.
