The algebraic degree of semidefinite programming
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 fun...
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| Veröffentlicht in: | Mathematical programming 2010-04, Vol.122 (2), p.379-405 |
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| container_title | Mathematical programming |
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| creator | Nie, Jiawang Ranestad, Kristian Sturmfels, Bernd |
| description | 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. |
| doi_str_mv | 10.1007/s10107-008-0253-6 |
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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. 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| subjects | Algebra Applied sciences Calculus of Variations and Optimal Control Optimization Codes Combinatorics Equilibrium Eulers equations Exact sciences and technology Full Length Paper Game theory Geometry Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Semidefinite programming Studies Theoretical |
| title | The algebraic degree of semidefinite programming |
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