The algebraic degree of semidefinite programming
نویسندگان
چکیده
منابع مشابه
The algebraic degree of semidefinite programming
Given a 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 met...
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We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting developments occurred in the last few years, including robust optimization, combinatorial optimizatio...
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In this note, we use a natural desingularization of the conormal variety of the variety of (n × n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming. 1. The algebraic degree in semidefinite programming Let P be a general projective space of symmetric (n×n)−matrices up to scalar multiples, and let Yr ⊂ P m be the subvariety of mat...
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In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many app...
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ژورنال
عنوان ژورنال: Mathematical Programming
سال: 2008
ISSN: 0025-5610,1436-4646
DOI: 10.1007/s10107-008-0253-6