Twice Differentiable Spectral Functions

Lewis, Adrian S.; Sendov, Hristo S.

doi:10.1137/s089547980036838x

Cited by 81 publications

(81 citation statements)

References 23 publications

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“…We first describe the basic method, which does not exploit sparsity of W. We first carry out Cholesky factorizations, (15) which also serves to verify that F and G are positive definite, which is equivalent to W − J < 1. We then compute the inverses of these Cholesky factors, and compute their Frobenius norms:…”

Section: Computing the Steady-state Mean-square Deviationmentioning

confidence: 99%

“…Furthermore, ss is twice continuously differentiable because the above symmetric function g is twice continuously differentiable at [y]. We can derive the gradient and Hessian of ss following the general formulas for spectral functions, as given in [4,Section 5.2,15]. In this paper, however, we derive simple expressions for the gradient and Hessian by directly applying the chain rule; see Section 3.…”

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

confidence: 99%

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Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

986

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: Computing the Steady-state Mean-square Deviationmentioning

confidence: 99%

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

986

732

View full text Add to dashboard Cite

show abstract

“…The differentiability result is established in [8, Theorem 1.1] and the twice differentiability in [9,Theorem 3.3]. The particular expression for the Hessian is given in [11,Example 4.9].…”

Section: Gradient and Hessian Of Spectral Functionsmentioning

confidence: 99%

Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices

Malick

Sendov

2006

Set-Valued Anal

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“…Analogues exist for C (∞) and analytic functions-see [19]. At the opposite extreme, at least for spectral functions, we have the following result (see [37,46]). …”

Section: Theorem 56 (Subdifferentials Of Invariant Functions) In Thmentioning

confidence: 64%

The mathematics of eigenvalue optimization

Lewis

2003

Math. Program., Ser. B

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Abstract. Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.Key words. Eigenvalue optimization -convexity -nonsmooth analysis -duality -semidefinite program -subdifferential -Clarke regular -chain rule -sensitivity -eigenvalue perturbation -partly smooth -spectral function -unitarily invariant norm -hyperbolic polynomial -stability -robust control -pseudospectrum -H∞ norm PART I: INTRODUCTION Von Neumann and invariant matrix normsBefore outlining this survey, I would like to suggest its flavour with a celebrated classical result. This result, von Neumann's characterization of unitarily invariant matrix norms, serves both as a historical jumping-off point and as an elegant juxtaposition of the central ingredients of this article.Von Neumann [64] was interested in unitarily invariant norms · on the vector space M n of n-by-n complex matrices:where U n denotes the group of unitary matrices. The singular value decomposition shows that the invariants of a matrix X under unitary transformations of the form X → U XV are given by the singular values σ 1 (X) ≥ σ 2 (X) ≥ · · · ≥ σ n (X), the eigenvalues of the matrix √ X * X. Hence any invariant norm · must be a function of the vector σ(X), so we can write X = g(σ(X)) for some function g : R n → R. We ensure this last equation if we define g(x) = Diag x for vectors x ∈ R n , and in that case g is a symmetric gauge function: that is, g is a norm on R n whose value is invariant under permutations and sign changes of the components. Von Neumann's beautiful insight was that this simple necessary condition is in fact also sufficient.Theorem 1.1 (von Neumann, 1937). The unitarily invariant matrix norms are exactly the symmetric gauge functions of the singular values.Von Neumann's proof rivals the result in elegance. He proceeds by calculating the dual norm of a matrix Y ∈ M n ,where the real inner product X, Y in M n is the real part of the trace of X * Y . Specifically, he shows that any symmetric gauge function g satisfies the simple duality relationshipwhere g * is the norm on R n dual to g. From this, the hard part of his result follows:...

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Twice Differentiable Spectral Functions

Cited by 81 publications

References 23 publications

Distributed average consensus with least-mean-square deviation

Distributed average consensus with least-mean-square deviation

Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices

The mathematics of eigenvalue optimization

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