Dissolving Constraints for Riemannian Optimization (Xin Liu)

　　In this paper, we consider optimization problems over closed embedded submanifolds of \mathbb{R}^n, 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 named 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 ？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.

　　Publication:

　　Mathematics of Operations Research, Published Online: 8 Mar 2023, https://doi.org/10.1287/moor.2023.1360

　　Author:

　　Nachuan Xiao

　　The Institute of Operations Research and Analytics, National University of Singapore, Singapore

　　Xin Liu

　　State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China

　　Email: liuxin@lsec.cc.ac.cn

　　Kim-Chuan Toh

　　Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, Singapore