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In this paper, we propose AdaBB, an adaptive gradient method based on the Barzilai-Borwein stepsize. The algorithm is line-search-free and parameter-free, and it essentially provides a convergent variant of the Barzilai-Borwein method for general convex optimization problems. We analyze the ergodic convergence of the objective function value and the convergence of the iterates for solving general convex optimization problems. Compared with existing works along this line of research, our algorithm gives the best lower bounds on the stepsize and the average of the stepsizes. Furthermore, we present extensions of the proposed algorithm for solving locally strongly convex and composite convex optimization problems where the objective function is the sum of a smooth function and a nonsmooth function. In the case of local strong convexity, we achieve linear convergence. Our numerical results also demonstrate very promising potential of the proposed algorithms on some representative examples. Funding: S. Ma is supported by the National Science Foundation [Grants DMS-2243650, CCF-2308597, CCF-2311275, and ECCS-2326591] and a startup fund from Rice University. J. Yang is supported by the National Natural Science Foundation of China [Grants 12431011 and 12371301] and the Natural Science Foundation for Distinguished Young Scholars of Gansu Province [Grant 22JR5RA223]. 

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function considered in the ambient space. This class of problems finds important applications in machine learning and statistics, such as sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an ϵ-stationary point is analyzed under mild assumptions. Existing ADMMs for solving nonconvex problems either do not allow a nonconvex constraint set or do not allow a nonsmooth objective function. Our algorithm is the first ADMM-type algorithm that minimizes a nonsmooth objective over manifold—a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method. Funding: The research of S. Ma was supported in part by the Office of Naval Research [Grant N00014-24-1-2705]; the National Science Foundation [Grants DMS-2243650, CCF-2308597, CCF-2311275, and ECCS-2326591]; the University of California, Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program; and Rice University start-up fund. 

In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded submanifolds of [Formula: see text]. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of [Formula: see text]. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. We also prove that any accumulation point of the sequence generated by our method that satisfies the Riemannian gradient subconsistency is a limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the advantages of Riemannian [Formula: see text] [Formula: see text] optimization over Riemannian [Formula: see text] optimization for finding sparse solutions and the effectiveness of the proposed method. Funding: C. Zhang was supported in part by the National Natural Science Foundation of China [Grant 12171027] and the Natural Science Foundation of Beijing [Grant 1202021]. X. Chen was supported in part by the Hong Kong Research Council [Grant PolyU15300219]. S. Ma was supported in part by the National Science Foundation [Grants DMS-2243650 and CCF-2308597], the UC Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program, and a startup fund from Rice University.