Hidden Convexity, Optimization, and Algorithms on Rotation Matrices

Akshay Ramachandran (), Kevin Shu () and Alex L. Wang ()
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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

Mathematics of Operations Research, 2025, vol. 50, issue 2, 1454-1477

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.

Keywords: Primary: 90C22; 90C26; Secondary: 70G65; 52A41; hidden convexity; special orthogonal group; Wahba’s problem (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:2:p:1454-1477

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