# Dissolving Constraints for Riemannian Optimization

arXiv: https://arxiv.org/abs/2203.10319 · HTML: https://mlsanity.com/paper/2203.10319/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2022.

## Built on by (most cited first, 12 of 12)

- [Guiding Neural Collapse: Optimising Towards the Nearest Simplex Equiangular Tight Frame](https://mlsanity.com/paper/2411.01248/) (2024)
- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2307.10053/) (2023)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach](https://mlsanity.com/paper/2507.22855/) (2025)
- [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026)
- [First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods](https://mlsanity.com/paper/2307.07605/) (2023)
- [A Double Tracking Method for Optimization with Decentralized Generalized Orthogonality Constraints](https://mlsanity.com/paper/2409.04998/) (2024)
- [A second-order method landing on the Stiefel manifold via Newtonunicodex2013Schulz iteration](https://mlsanity.com/paper/2605.02838/) (2026)
- [Retraction-Free Optimization over the Stiefel Manifold for the LoRA Fine-Tuning](https://mlsanity.com/paper/2607.25299/) (2026)

## Abstract (arXiv)

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 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 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.