Distance Optimization and the Extremal Variety of the Grassmann Variety

John Leventides (), George Petroulakis () and Nicos Karcanias ()
Additional contact information
John Leventides: University of Athens
George Petroulakis: City University London
Nicos Karcanias: City University London

Journal of Optimization Theory and Applications, 2016, vol. 169, issue 1, No 1, 16 pages

Abstract: Abstract The approximation of a multivector by a decomposable one is a distance-optimization problem between the multivector and the Grassmann variety of lines in a projective space. When the multivector diverges from the Grassmann variety, then the approximate solution sought is the worst possible. In this paper, it is shown that the worst solution of this problem is achieved, when the eigenvalues of the matrix representation of a related two-vector are all equal. Then, all these pathological points form a projective variety. We derive the equation describing this projective variety, as well as its maximum distance from the corresponding Grassmann variety. Several geometric and algebraic properties of this extremal variety are examined, providing a new aspect for the Grassmann varieties and the respective projective spaces.

Keywords: Distance geometry problems; Optimization; Approximations; Projective varieties; Sums of squares and representations; 78M50; 57Q55; 08B30; 11E25 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-015-0840-7

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