Global optimization on matrix Lie groups via intermittent diffusion: A geometric stochastic framework with provable convergence

This paper proposes a stochastic differential equation (SDE)-based global optimization method on matrix Lie groups, specifically addressing the challenge of local optima trapping in optimization over special orthogonal groups SO(n) and related structures. By combining Riemannian conjugate gradient methods with an intermittent diffusion mechanism, we develop an optimization framework that preserves geometric structures while maintaining global exploration capabilities. Theoretical analysis demonstrates first-order convergence between the continuous-time SDE and its discrete numerical implementation, with almost sure convergence to the global optimum under appropriate conditions. Experimental validation through two applications—multimodal function optimization and robotic path planning—confirms the effectiveness of the proposed ID-RCG algorithm, showing superior performance in both convergence rate and global search capability compared to conventional methods.