Some geometric results in semidefinite programming

Ramana, Motakuri V.; Goldman, A. J.

doi:10.1007/bf01100204

Cited by 139 publications

(102 citation statements)

References 22 publications

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“…Results on the image of quadratic mappings play an important role in nonconvex quadratic optimization (see, e.g., [17,25,27] and references therein). We begin with the following theorem due to Polyak [25, Theorem 2.2], which is very relevant to our analysis.…”

Section: The Image Of the Complex And Real Space Under A Quadratic Mamentioning

confidence: 99%

Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints

Beck¹,

Eldar²

2006

SIAM J. Optim.

214

146

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Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.

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Section: The Image Of the Complex And Real Space Under A Quadratic Mamentioning

confidence: 99%

Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints

Beck¹,

Eldar²

2006

SIAM J. Optim.

214

146

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“…Independently, Ramana and Goldman (1995) derived several results on the geometry of the feasible sets of SDPs (such as a characterization of faces).…”

Section: Is the Total Number Of Equality Constraints If A Constraintmentioning

confidence: 99%

On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues

1998

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We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalue-optimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrix-valued function.We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue-optimization. In the spectrum of an optimal matrix, the kth and (k / 1)st largest eigenvalues tend to be equal and frequently have multiplicity greater than two. This clustering is intuitively plausible and has been observed as early as 1975.When the matrix-valued function is affine, we prove that clustering must occur at extreme points of the set of optimal solutions, if the number of variables is sufficiently large. We also give a lower bound on the multiplicity of the critical eigenvalue. These results generalize to the case of a general matrix-valued function under appropriate conditions.

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“…This result in Remark 5.43 is due to Ramana and Goldman [30]. In summary, while the dual to a spectrahedron is generally not a spectrahedron, it is always a projected spectrahedron.…”

Section: Spectrahedra and Semidefinite Programmingmentioning

confidence: 58%