J.B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numer. Math., 70:45-72, 1995. Home/Search Document Details and Download
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Twice Differentiable Spectral Functions - Lewis, Sendov (2001) (Correct)
....B and the vector b depend on the point and the function f . Lemma 2.3 For 2 R n # and a sequence of symmetric matrices Mm 0 we have that (Diag Mm ) T = T (X T 1 MmX 1 ) T ; X T r MmX r ) T o(kMm k) Proof. Combine Lemma 5.10 in [10] and Theorem 3. 12 in [3]. The following is our main technical tool. Lemma 2.4 Let fMmg be a sequence of symmetric matrices converging to 0, such that Mm =kMm k converges to M . Let be in R n # and Um U 2 O n be a sequence of orthogonal matrices such that Diag Mm = Um Diag (Diag Mm ) U T m ; for ....
J.-B. HIRIART-URRUTY and D. YE. Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Mathematik, 70:45-72, 1992.
Quadratic Expansions of Spectral Functions - Lewis, Sendov (2000) (Correct)
.... k l , 1 l r. We denote the standard basis in R n by e 1 ,e 2 , e n . As a byproduct in the following lemma we derive a formula for the derivative of the function k l at the point Diag . This formula appeared many times in the literature: see 8 for example Corollary 3. 10 in [5], or the proof of Corollary 3.3 in [9] The expression for the Hessian is also known, see Formula (3.28) in [13] here we present yet another way of deriving it. Lemma 4.5 For a real vector 2 R n , such that 1 = k 1 k 1 1 = k 2 k 2 1 kr ; k 0 = 0; k r ....
J.-B. HIRIART-URRUTY and D. YE. Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Mathematik, 70:45-72, 1992.
Group Invariance and Convex Matrix Analysis - Lewis (1995) (5 citations) (Correct)
....yet powerful unifying framework in which to study a wide variety of important results. Examples include Schur convexity (see for example [22] the convexity of eigenvalue functions ( 10, 6, 11, 3, 13, 19] calculations of Fenchel conjugates and subdifferentials of convex eigenvalue functions [24, 5, 12, 30, 28, 25, 26, 27, 15, 16, 1, 17, 19], von Neumann s original result [33] and generalizations (for example [4, 20] subdifferentials of unitarily invariant norms [34, 35, 36, 37, 38, 8, 7, 9, 20] and characterizations of extreme, exposed and smooth points of unit balls [2, 37, 38, 8, 7, 9, 20] This paper concentrates on convexity ....
....in [19] The latter paper also contains the remainder of Theorem 8.1. A proof appears in [32] that h ffi is analytic at x if and only if h is analytic at (x) some somewhat related results appear in [21] Numerous formulae for subgradients of specific matrix functions appear, for example, in [26, 27, 15, 16]: the chain rule in Theorem 8.1 provides a simple unified approach to these. Theorem 8.4 (Spectral convex sets) Weakly orthogonally invariant subsets of S n are exactly those sets of the form Gamma1 (C) for symmetric subsets C of R n , If the symmetric matrix x has (x) in the symmetric set ....
J.-B. Hiriart-Urruty and D. Ye. Sensitivity analysis of all eigenvalues of a symmetric matrix. Technical report, Laboratoire d'analyse numerique, Universit' e Paul Sabatier, Toulouse, France, 1992.
Semidefinite Programming - Vandenberghe, Boyd (1994) (248 citations) (Correct)
.... which can be cast as a semidefinite program (see x2) See, for instance, Cullum, Donath and Wolfe [CDW75] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92] Overton and Womersley [OW93, OW92] Ringertz [Rin91] Fan and Nekooie [FN92] Fan [Fan93] Hiriart Urruty and Ye [HUY95], Shapiro and Fan [SF94] and Pataki [Pat94] Interior point methods for LPs were introduced by Karmarkar in 1984 [Kar84] although many of the underlying principles are older (see, e.g. Fiacco and McCormick [FM68] Lieu and Huard [LH66] and Dikin [Dik67] Karmarkar s algorithm, and the ....
J.-B. Hiriart-Urruty and D. Ye. Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Mathematik, 70:45--72, 1995.
Twice Dierentiable Spectral Functions - Adrian Lewis And (Correct)
....the vector b depend on the point and the function f . Lemma 2.3 For # R n # and a sequence of symmetric matrices Mm # 0 we have that #(Diag Mm ) T = T # #(X T 1 MmX 1 ) T , #(X T r MmX r ) T # o(#Mm #) 4 Proof. Combine Lemma 5.10 in [8] and Theorem 3. 12 in [1]. # The following is our main technical tool. Lemma 2.4 Let Mm be a sequence of symmetric matrices converging to 0, such that Mm #Mm # converges to M . Let be in R n # and Um # U # O n be a sequence of orthogonal matrices such that Diag Mm = Um # Diag #(Diag Mm ) # U T ....
J.-B. HIRIART-URRUTY and D. YE. Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Mathematik, 70:45--72, 1992.
Derivatives Of Orbital Functions, An Extension Of.. - Tam, Hill (Correct)
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J.B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numer. Math., 70:45-72, 1995.
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