A swarm intelligence optimization algorithm on riemannian manifolds

Riemannian manifold optimization methods solve constrained optimization problems, by transforming them into unconstrained problems on the Riemannian manifold. They can fully utilize the intrinsic geometric structure of the problem so that efficiency and performance of solution can be enhanced significantly. However, most of the Riemannian manifold optimization methods are local search approaches and requires the objective functions to be differentiable, and therefore many metaheuristic methods have been adapted to the Riemannian manifolds. In this paper, based on the quantum-behaved particle swarm optimization, we proposed a so-called Riemannian adapted quantum-behaved particle swarm optimization (RAQPSO), for the purpose of taking the advantage of the global search ability of the QPSO to address the local restriction encountered in the existing Riemannian manifold optimization methods. The RAQPSO operates within the retraction framework, projecting particles onto the tangent space of the manifold for updates and retracting them back to the manifold. In order to better adapt the QPSO algorithm to the manifold, we used two-stage mean projection method and Gaussian-based method respectively to calculate mean optimal position and local attractor points. The algorithm exhibited state-of-the-art performance on four benchmark problems posed on different kinds of matrix manifolds and had been applied to the MIMO system.