arXiv:2605.22736v1 Announce Type: cross Abstract: Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities -- clean intersection and intrinsic transversality -- are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and
The paper, published in 2026, represents a fundamental advance in optimization theory, a core component of many AI and scientific computing algorithms.
Improved optimization techniques can significantly enhance the efficiency and capability of AI models and complex engineering solutions, leading to faster research cycles and more effective applications.
This research provides a more tractable method for solving previously challenging optimization problems involving multiple geometric constraints, potentially unlocking new computational pathways.
- · AI/ML researchers
- · Scientific computing sector
- · Optimization software developers
- · Robotics engineers
More efficient training of machine learning models and complex AI systems.
Accelerated development of AI agents capable of navigating complex, constrained environments.
The ability to tackle previously intractable engineering and design challenges in fields like materials science or drug discovery.
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