Simple Algorithms for Optimization on Riemannian Manifolds with Constraints

Changshuo Liu, Nicolas Boumal

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions.

Original languageEnglish (US)
Pages (from-to)949-981
Number of pages33
JournalApplied Mathematics and Optimization
Issue number3
StatePublished - Dec 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics


  • Augmented Lagrangian method
  • Constrained optimization
  • Differential geometry
  • Exact penalty method
  • Nonsmooth optimization
  • Riemannian optimization


Dive into the research topics of 'Simple Algorithms for Optimization on Riemannian Manifolds with Constraints'. Together they form a unique fingerprint.

Cite this