An improved proximal alternating linearized minimization algorithm and its inertial Bregman extension for structured nonconvex nonsmooth optimization problems on Riemannian manifold

In this paper, we study a class of nonconvex and nonsmooth optimization problems with a specific structure. The objective function consists of the following components: a nonsmooth function in Euclidean space, a smooth function on a manifold, and a smooth coupling function linking of these two variables. By exploiting problem’s composite structure, we propose an improved version of the proximal alternating linearized minimization method. This algorithm employs a splitting technique, where the nonsmooth subproblem defined in the Euclidean space is solved using a forward–backward operator, and the smooth manifold subproblem is addressed using a Riemannian gradient step. We further extend this algorithm by introducing an inertial Bregman variant, which incorporates inertial step and Bregman regularization to improve both efficiency and robustness. We provide a theoretical analysis of the proposed algorithms and establish their iteration complexity for obtaining an $ϵ$-stationary point under mild assumptions. Finally, we present numerical experiments on sparse principal component analysis to verify the effectiveness of the proposed algorithms.