Dissolving Constraints for Riemannian Optimization
提出一类约束消解方法，将黎曼优化问题转化为无约束优化问题，并证明两者在可行域邻域内具有相同的一阶和二阶稳定点、局部极小点及Łojasiewicz指数，从而直接利用无约束优化的收敛性结果。
In this paper, we consider optimization problems over closed embedded submanifolds of [Formula: see text], 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 Łojasiewicz 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. Funding: The research of N. Xiao and K.-C. Toh is supported by the Ministry of Education of Singapore Academic Research Fund Tier 3 [Grant MOE-2019-T3-1-010]. The research of X. Liu is supported in part by the National Natural Science Foundation of China [Grants 12125108, 11971466, 12288201, 12021001, and 11991021]; the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711904]; and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences [Grant ZDBS-LY-7022].