Leading eigenvalue problems for large scale matrices arise in many applications. Coordinate-wise descent methods are considered in this talk for such problems based on a reformulation of the leading eigenvalue problem as a nonconvex optimization problem. The convergence of several coordinate-wise methods, including a randomized coordinate-wise method, is analyzed and compared. Numerical examples of applications to quantum many-body problems demonstrate the efficiency and provide benchmarks of the proposed coordinate-wise descent methods.

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