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In this talk, we consider optimization problems over closed embedded submanifolds of Euclidean space, which are defined by the equality constraints c(x)=0. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. Its main idea is to transfer the original manifold constrained optimization into an unconstrained optimization problem which minimizes a constraint dissolving function abbreviated as CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF share the same first-order and second-order stationary points, local minimizers, and ?ojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches.
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