In this paper, we propose the Manifold Random Drift Particle Swarm Optimization (MRDPSO) algorithm for matrix optimization on smooth manifolds. Conventional swarm intelligence methods generally converge prematurely in constrained domains. To mitigate this issue, we introduce the swarm intelligence methods to the manifold and a Random Drift mechanism that regulates the search process. Using Riemannian geometry, our framework treats constrained problems as unconstrained ones on the manifold, which preserves the intrinsic geometric structure of the data. Particles are initialized on the manifold, while updates are performed in tangent spaces. Since geodesic calculations are computationally expensive, we use an inverse retraction as a faster alternative to standard logarithmic mapping. Numerical experiments on Grassmann, Stiefel, and Oblique manifolds show that MRDPSO achieves higher accuracy and superior convergence stability compared to recent state-of-the-art manifold-adapted heuristics, namely IISSO and MSSO.
Halimu et al. (Mon,) studied this question.