Second-order directional derivatives of all eigenvalues of a symmetric matrix

Torki, Mounir

doi:10.1016/s0362-546x(00)00165-6

Cited by 38 publications

(25 citation statements)

References 10 publications

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“…From the above lemma, we get a perturbation result about eigenvalues λ i (G + G) for i ∈ β, which plays a key role in developing the formula for the directional derivatives whose proof is quite similar to that of [10,Proposition 1.4]. …”

Section: Lemma 23mentioning

confidence: 95%

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

Zhang

Pang

2011

J Optim Theory Appl

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This paper focuses on the study of differential properties of the symmetric matrix-valued Fischer-Burmeister (FB) function. As the main results, the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian of the symmetric matrix-valued Fischer-Burmeister function are established, which can be utilized in designing implementable Newton-type algorithms for nonsmooth equations involving the symmetric matrix-valued FB function.

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Section: Lemma 23mentioning

confidence: 95%

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

Zhang

Pang

2011

J Optim Theory Appl

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“…. , A n , respectively, as proved in [17] and considered as sub-gradient in [16] [18]. In our problem, A = D 1/2 PD −1/2 − √ π √ π T .…”

Section: For a Symmetric Matrix A(x) =mentioning

confidence: 98%

Fast Markov Decision Process for data collection in sensor networks

Duong

Nguyen

2014

2014 23rd International Conference on Computer Communication and Networks (ICCCN)

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We investigate the data collection problem in sensor networks. The network consists of a number of stationary sensors deployed at different sites for sensing and storing data locally. A mobile element moves from sites to sites to collect data from the sensors periodically. There are different costs associated with the mobile element moving from one site to another, and different rewards for obtaining data at different sensors. Furthermore, the costs and the rewards are assumed to change abruptly. The goal is to find a "fast" optimal movement pattern/policy of the mobile element that optimizes for the costs and rewards in non-stationary environments. We propose a novel optimization framework called Fast Markov Decision Process (FMDP) to solve this problem. The proposed FMDP framework extends the classical Markov Decision Process theory by incorporating the notion of mixing time that allows for the tradeoff between the optimality and the convergence rate to the optimality of a policy. Theoretical and simulation results are provided to verify the proposed approach.

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“…It is well known since 7 that is C C in fact and one can my i m w x derive an expression of its second-order differential from 24 …”

Section: The Eigenvaluementioning

confidence: 99%

First- and Second-Order Epi-Differentiability in Eigenvalue Optimization

Torki¹

1999

Journal of Mathematical Analysis and Applications

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The subject of this work is the first-and second-order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi-differentiability. We show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. We also prove that the mth largest eigenvalue function is twice epi-differentiable if and only if it ranks first in a group of equal eigenvalues. Finally, we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems. ᮊ 1999 Academic Press

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Second-order directional derivatives of all eigenvalues of a symmetric matrix

Cited by 38 publications

References 10 publications

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

Fast Markov Decision Process for data collection in sensor networks

First- and Second-Order Epi-Differentiability in Eigenvalue Optimization

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