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Hidden Convexity, Optimization, and Algorithms on Rotation Matrices

Author

Listed:
  • Akshay Ramachandran

    (Centrum Wiskunde & Informatica, 1098 XG Amsterdam, Netherlands)

  • Kevin Shu

    (Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Alex L. Wang

    (Centrum Wiskunde & Informatica, 1098 XG Amsterdam, Netherlands; and Purdue University, West Lafayette, Indiana 47907)

Abstract

This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices SO ( n ) . Such problems are nonconvex because of the constraint X ∈ SO ( n ) . Nonetheless, we show that certain linear images of SO ( n ) are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of SO ( n ) is convex and that the projection of SO ( n ) onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over SO ( n ) with a single constraint or with constraints defined by low-rank matrices. Both of these results are maximal in a formal sense.

Suggested Citation

  • Akshay Ramachandran & Kevin Shu & Alex L. Wang, 2025. "Hidden Convexity, Optimization, and Algorithms on Rotation Matrices," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 1454-1477, May.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:2:p:1454-1477
    DOI: 10.1287/moor.2023.0114
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