Journal
MATHEMATICAL PROGRAMMING
Volume 122, Issue 2, Pages 379-405Publisher
SPRINGER
DOI: 10.1007/s10107-008-0253-6
Keywords
Semidefinite programming; Algebraic degree; Genericity; Determinantal variety; Dual variety; Multidegree; Euler-Poincare characteristic; Chern class
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Funding
- US National Science Foundation [DMS-0456960]
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Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.
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