An Efficient Algorithm for Computing the Generalized Null Space Decomposition

Guglielmi, Nicola; Overton, Michael L.; Stewart, G. W.

doi:10.1137/140956737

“…We found that almost all optimization runs generated function values converging either to the conjectured globally minimal value 0.5, or to the evidently locally minimal value 1. In the former case, making use of the generalized null space decomposition [13], we confirmed that the computed final (p, A) always approximated the conjectured globally minimal "Crabb matrix" configurations, to be described below, with field of values a circular disk. In the latter case, using the Schur factorization, we confirmed that the final (p, A) always approximated an "ice-cream-cone" configuration, again to be described below.…”

Section: Introductionsupporting

confidence: 72%

Local minimizers of the Crouzeix ratio: a nonsmooth optimization case study

Overton

¹

2021

Calcolo

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Given a square matrix A and a polynomial p, the Crouzeix ratio is the norm of the polynomial on the field of values of A divided by the 2-norm of the matrix p(A). Crouzeix's conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occurring locally minimal value. Making use of a heavy-tailed distribution to initialize our optimization computations, we demonstrate for the first time that the Crouzeix ratio has many other locally minimal values between 0.5 and 1. Besides showing that the same function values are repeatedly obtained for many different starting points, we also verify that an approximate nonsmooth stationarity condition holds at computed candidate local minimizers. We also find that the same locally minimal values are often obtained both when optimizing over real matrices and polynomials, and over complex matrices and polynomials. We argue that minimization of the Crouzeix ratio makes a very interesting nonsmooth optimization case study, illustrating among other things how effective the BFGS method is for nonsmooth, nonconvex optimization. Our method for verifying approximate nonsmooth stationarity is based on what may be a novel approach to finding approximate subgradients of max functions on an interval. Our extensive computations strongly support Crouzeix's conjecture: in all cases, we find that the smallest locally minimal value is 0.5.

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“…We found that almost all optimization runs generated function values converging either to the conjectured globally minimal value 0.5, or to the evidently locally minimal value 1. In the former case, making use of the generalized null space decomposition [GOS15], we confirmed that the computed final (p, A) always approximated the conjectured globally minimal "Crabb matrix" configurations, to be described below, with field of values a circular disk. In the latter case, using the Schur factorization, we confirmed that the final (p, A) always approximated an "ice-cream-cone" configuration, again to be described below.…”

Section: Introductionsupporting

confidence: 72%

Local Minimizers of the Crouzeix Ratio: A Nonsmooth Optimization Case Study

Overton¹

2021

Preprint

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0

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Given a square matrix A and a polynomial p, the Crouzeix ratio is the norm of the polynomial on the field of values of A divided by the 2-norm of the matrix p(A). Crouzeix's conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occurring locally minimal value. Making use of a heavy-tailed distribution to initialize our optimization computations, we demonstrate for the first time that the Crouzeix ratio has many other locally minimal values between 0.5 and 1. Besides showing that the same function values are repeatedly obtained for many different starting points, we also verify that an approximate nonsmooth stationarity condition holds at computed candidate local minimizers. We also find that the same locally minimal values are often obtained both when optimizing over real matrices and polynomials, and over complex matrices and polynomials. We argue that minimization of the Crouzeix ratio makes a very interesting nonsmooth optimization case study, illustrating among other things how effective the BFGS method is for nonsmooth, nonconvex optimization. Our method for verifying approximate nonsmooth stationarity is based on what may be a novel approach to finding approximate subgradients of max functions on an interval. Our extensive computations strongly support Crouzeix's conjecture: in all cases, we find that the smallest locally minimal value is 0.5.

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“…Let the system of (32) be regular. From Lemma 4, it follows that (32) can be rewritten as [ 37 , 38 ]:

where

;

; and

.…”

Section: The General Solution Of Folti Singular Systems With Regulmentioning

confidence: 99%

The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

Batiha

¹

,

El-Khazali

²

,

Alsaedi

³

et al. 2018

Entropy

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This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, which depends on introducing a method of decomposing their dynamic states from their static states. The solution is formulated by converting the singular system of regular pencils into a recursive form using the sequence of transformations, which separates the dynamic variables from the algebraic variables. The main idea of this work is demonstrated via numerical examples.

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An Efficient Algorithm for Computing the Generalized Null Space Decomposition

Cited by 21 publications

References 17 publications

Local minimizers of the Crouzeix ratio: a nonsmooth optimization case study

Local minimizers of the Crouzeix ratio: a nonsmooth optimization case study

Local Minimizers of the Crouzeix Ratio: A Nonsmooth Optimization Case Study

The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

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