Distance Optimization and the Extremal Variety of the Grassmann Variety

Leventides, J., Petroulakis, G. & Karcanias, N. (2016). Distance Optimization and the Extremal Variety of the Grassmann Variety. Journal of Optimization Theory and Applications, 169(1), pp. 1-16. doi: 10.1007/s10957-015-0840-7

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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.

Item Type: Article
Additional Information: The final publication will be available at Springer http://www.springer.com/ on publication.
Uncontrolled Keywords: Distance geometry problems, Optimization, Approximations, Projective varieties, Sums of squares and representations
Subjects: Q Science > QA Mathematics
Divisions: School of Engineering & Mathematical Sciences
URI: http://openaccess.city.ac.uk/id/eprint/13343

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