Chapter 5: Dualities

Rostalski, Philipp; Sturmfels, Bernd

doi:10.1137/1.9781611972290.ch5

Cited by 9 publications

(8 citation statements)

References 25 publications

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“…Namely, Q is a generic spectrahedron of codimension p + 1 in the space of symmetric n × n matrices. By [10,Theorem 5.50], its extreme points are matrices of rank n − r for some r that lies in the Pataki range (1). We see each one of these ranks as the rank of extreme points with a positive probability.…”

Section: One Proof and Many Numbersmentioning

confidence: 99%

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Sinn¹,

Sturmfels²

2015

SIAM J. Optim.

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Section: One Proof and Many Numbersmentioning

confidence: 99%

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Sinn¹,

Sturmfels²

2015

SIAM J. Optim.

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“…Yalmip is primarily a wrapper for various optimization problems in MATLAB [Löfberg, 2004], but it has an undocumented extension for noncommuting variables. Bermeja, a convex algebraic geometry package builds on this undocumented feature to solve NC problems [Rostalski, 2012]. Unfortunately it does not scale beyond a few noncommuting variables, and in the most recent releases of Yalmip, the noncommuting variables are being phased out.…”

Section: Related Workmentioning

confidence: 99%

Algorithm 950

Wittek

2015

ACM Trans. Math. Softw.

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A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting problems that creates the relaxation to be solved by SDPA -a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve problems such as finding the ground state energy or testing quantum correlations.

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“…By passing to the Zariski closures, we conclude that the algebraic boundaries of Σ 3,6 \P 3,6 and Σ 4,4 \P 4,4 are projectively dual to the Hankel determinantal varieties above. For a general introduction to the relationship between projective duality and cone duality in convex algebraic geometry we refer to [25].…”

Section: Rank Conditions On Hankel Matricesmentioning

confidence: 99%

Algebraic boundaries of Hilbert’s SOS cones

et al. 2012

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Abstract. We study the geometry underlying the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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Chapter 5: Dualities

Cited by 9 publications

References 25 publications

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Algebraic boundaries of Hilbert’s SOS cones

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