Dissolving Constraints for Riemannian Optimization

In this paper, we consider optimization problems over closed embedded submanifolds of Rn, which are defined by the constraints c(x) = 0. We propose a class of constraint-dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint-dissolving function (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 have the same first-order and second-order stationary points, local minimizers, and Lojasiewicz 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.