# Simple algorithms for optimization on Riemannian manifolds with constraints

arXiv: https://arxiv.org/abs/1901.10000 · HTML: https://mlsanity.com/paper/1901.10000/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2019.

## Built on by (most cited first, 14 of 14)

- [Fast and Efficient MMD-based Fair PCA via Optimization over Stiefel Manifold](https://mlsanity.com/paper/2109.11196/) (2021)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size](https://mlsanity.com/paper/2501.18164/) (2025)
- [Invex Programs: First Order Algorithms and Their Convergence](https://mlsanity.com/paper/2307.04456/) (2023)
- [Structured Regularization for Constrained Optimization on the SPD Manifold](https://mlsanity.com/paper/2410.09660/) (2024)
- [Suboptimality bounds for trace-bounded SDPs enable a faster and scalable low-rank SDP solver SDPLR+](https://mlsanity.com/paper/2406.10407/) (2024)
- [Riemannian Geometric-based Meta Learning](https://mlsanity.com/paper/2503.10993/) (2025)
- [A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints](https://mlsanity.com/paper/2511.03443/) (2025)
- [Robust Least-Squares Optimization for Data-Driven Predictive Control: A Geometric Approach](https://mlsanity.com/paper/2511.09242/) (2025)
- [Information Theoretic Bayesian Optimization over the Probability Simplex](https://mlsanity.com/paper/2603.09793/) (2026)
- [Deflation-Free Optimal Scoring](https://mlsanity.com/paper/2604.25664/) (2026)
- [Optimization over the intersection of manifolds](https://mlsanity.com/paper/2605.22736/) (2026)
- [Beyond Additive Decompositions: Interpretability Through Separability](https://mlsanity.com/paper/2605.31200/) (2026)

## Abstract (arXiv)

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.