An inertial Riemannian gradient ADMM for nonsmooth manifold optimization

\(\) The Alternating Direction Method of Multipliers (ADMM) is widely recognized for its efficiency in solving separable optimization problems. However, its application to optimization on Riemannian manifolds remains a significant challenge. In this paper, we propose a novel inertial Riemannian gradient ADMM (iRG-ADMM) to solve Riemannian optimization problems with nonlinear constraints. Our key contributions are as follows: (i) we introduce an inertial strategy applied to the Riemannian gradient, enabling faster convergence for smooth subproblems constrained on Riemannian manifolds; (ii) for nonsmooth subproblems in Euclidean space, we incorporate existing well-established algorithms for efficient solution; and (iii) we establish the $\epsilon$-stationarity of iRG-ADMM under mild conditions. Finally, we demonstrate the effectiveness of iRG-ADMM through extensive numerical experiments, including applications to Sparse Principal Component Analysis (SPCA), highlighting its superior performance compared to existing methods.

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