Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

Overton, Michael L.; Womersley, Robert S.

doi:10.1007/bf01585173

Cited by 157 publications

(93 citation statements)

References 16 publications

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“…Examples of convex spectral functions include the trace, largest eigenvalue, and the sum of the k largest eigenvalues, for any symmetric matrix; and the trace of the inverse, and log determinant of the inverse, for any positive definite matrix. More examples and details can be found in, e.g., [14,23].…”

Section: (G • )(Qw Q T ) = (G • )(W )mentioning

confidence: 99%

Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

975

732

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We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. In this model, each node updates its local variable with a weighted average of its neighbors' values, and each new value is corrupted by an additive noise with zero mean. The quality of consensus can be measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. We consider the problem of finding the (symmetric) edge weights that result in the least mean-square deviation in steady state. We show that this problem can be cast as a convex optimization problem, so the global solution can be found efficiently. We describe some computational methods for solving this problem, and compare the weights and the mean-square deviations obtained by this method and several other weight design methods.

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Section: (G • )(Qw Q T ) = (G • )(W )mentioning

confidence: 99%

Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

975

732

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“…The example (23) is perhaps the most interesting one since it appears in concrete problems of optimization [12] and principal components analysis [14].…”

Section: Lemma 3 One Hasmentioning

confidence: 99%

Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory

Iusem¹,

Seeger²

2007

Mat. apl. comput.

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Abstract. There are three related concepts that arise in connection with the angular analysis of a convex cone: antipodality, criticality, and Nash equilibria. These concepts are geometric in nature but they can also be approached from the perspective of optimization theory. A detailed angular analysis of polyhedral convex cones has been carried out in a recent work of ours. This note focus on two important classes of non-polyhedral convex cones: elliptic cones in an Euclidean vector space and spectral cones in a space of symmetric matrices.Mathematical subject classification: 52A40, 90C26.

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“…Therefore constraints (2c) and (2d) can be relaxed into Tr(Z) = d and 0 Z I, which are both convex. When the cost function is linear and it is subject to Ω 2 , the solution will be at one of the extreme points [12]. Consequently, for linear cost functions, the optimization problems subject to Ω 1 and Ω 2 are exactly equivalent.…”

Section: Let Us Define a New Variablementioning

confidence: 99%

Supervised dimensionality reduction via sequential semidefinite programming

Shen

Brooks

2008

Pattern Recognition

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Cite this article as: Chunhua Shen, Hongdong Li and Michael J. Brooks, Supervised dimensionality reduction via sequential semidefinite programming, Pattern Recognition (2008Recognition ( ), doi:10.1016Recognition ( /j.patcog.2008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractMany dimensionality reduction problems end up with a trace quotient formulation. Since it is difficult to directly solve the trace quotient problem, traditionally the trace quotient cost function is replaced by an approximation such that generalized eigenvalue decomposition can be applied. In contrast, we directly optimize the trace quotient in this work. It is reformulated as a quasi-linear semidefinite optimization problem, which can be solved globally and efficiently using standard off-the-shelf semidefinite programming solvers. Also this optimization strategy allows one to enforce additional constraints (for example, sparseness constraints) on the projection matrix. We apply this optimization framework to a novel dimensionality reduction algorithm. The performance of the proposed algorithm is demonstrated in experiments on several UCI machine learning benchmark examples, USPS handwritten digits as well as ORL and Yale face data.

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Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

Cited by 157 publications

References 16 publications

Distributed average consensus with least-mean-square deviation

Distributed average consensus with least-mean-square deviation

Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory

Supervised dimensionality reduction via sequential semidefinite programming

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