The Algebraic Degree of Semidefinite Programming

Details The Algebraic Degree of Semidefinite Programming The Algebraic Degree of Semidefinite Programming

2006-11-19

arxiv.org/abs/math/0611562v3

Mathematics

Optimization and Control

23 pages, 3 figures

Scientific paper

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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