Authors: Dodig, Marija 
Stošić, Marko 
Xavier, João
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On minimizing a quadratic function on Stiefel manifold
Journal: Linear Algebra and Its Applications
Volume: 475
First page: 251
Last page: 264
Issue Date: 15-Jun-2015
Rank: M21
ISSN: 0024-3795
DOI: 10.1016/j.laa.2015.02.028
Abstract: 
In this paper we propose a novel approach to a particular quadratic programming problem, when the optimization is performed over the set O(3,2) of 3×2 Stiefel matrices. We rewrite the original nonconvex problem as a semi-definite programming problem, by computing a convex hull (tight convex relaxation) of a certain set of matrices. We give an efficient, quick algorithm for the minimization of a quadratic function over Stiefel manifold. We report some numerical experiments to illustrate the tightness of the convex approximation obtained by the two aforementioned methods ("standard" and ours). Our result is of immediate interest in Computer Vision, including Structure-from-Motion (SfM) problems, and 2D-3D registration.
Keywords: Convex hull | Quadratic programming | Semi-definite programming | Stiefel matrix
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Geometry, Education and Visualization With Applications 
FCT, Grants CMU-PT/SIA/0026/2009, PTDC/EMS-CRO/2042/2012 and UID/EEA/5009/2013

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