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  • 1
    Digitale Medien
    Digitale Medien
    Pseudozeros of polynomials and pseudospectra of companion matrices (1994)
    Springer
    Numerische Mathematik 68 (1994), S. 403-425 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991): 65F35
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. It is well known that the zeros of a polynomial $p$ are equal to the eigenvalues of the associated companion matrix $A$ . In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The $\epsilon$-$pseudozero \: set \: Z_{\epsilon}(p)$ is the set of zeros of all polynomials $\hat{p}$ obtained by coefficientwise perturbations of $p$ of size {$\leq \epsilon$} ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The $\epsilon$-$pseudospectrum \: \Lambda_\epsilon(A)$ is another subset of ${\Bbb C}$ defined as the set of eigenvalues of matrices {$\hat{A} = A + E$} with $\Vert E\Vert \leq \epsilon$ ; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then $Z_{\epsilon \Vert p\Vert }(p)$ and $\Lambda_{ \epsilon \Vert A\Vert }(A)$ are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050069
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  • 2
    Digitale Medien
    Digitale Medien
    Primal-Dual Path-Following Algorithms for Determinant Maximization Problems With Linear Matrix Inequalities (1999)
    Springer
    Computational optimization and applications 14 (1999), S. 309-330 
    Details
    ISSN: 1573-2894
    Schlagwort(e): determinant optimization ; semidefinite programming ; predictor-corrector
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik
    Notizen: Abstract Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton's method to a primal-dual central path equation similar to that in semidefinite programming (SDP) to obtain a Newton system which is then symmetrized to avoid nonsymmetric search direction. Computational aspects of the algorithms are discussed, including Mehrotra-type predictor-corrector variants. Focusing on three different symmetrizations, which leads to what are known as the AHO, H..K..M and NT directions in SDP, numerical results for various classes of maxdet-problem are given. The computational results show that the proposed algorithms are efficient, robust and accurate.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1026400522929
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  • 3
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    Solving Challenging Large Scale QAPs (2021)
    Details
    Publikationsdatum: 2021-12-09
    Beschreibung: We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method efficiently implemented on a powerful computer system using the Ubiquity Generator(UG) framework that can utilize more than 100,000 cores. Lower bounding procedures incorporated in the branch-and-bound method play a crucial role in solving the problems. For a strong lower bounding procedure, we employ the Lagrangian doubly nonnegative (DNN) relaxation and the Newton-bracketing method developed by the authors’ group. In this report, we describe some basic tools used in the project including the lower bounding procedure and branching rules, and present some preliminary numerical results. Our next target problem is QAPs with dimension at least 50, as we have succeeded to solve tai30a and sko42 from QAPLIB for the first time.
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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  • 4
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    大規模二次割当問題への挑戦 (2022)
    Details
    Publikationsdatum: 2022-05-13
    Beschreibung: 二次割当問題は線形緩和が弱いことが知られ，強化のため多様な緩和手法が考案されているが，その一つである二重非負値計画緩和（ DNN 緩和）及びその解法として近年研究が進んでいるニュートン・ブラケット法を紹介し，それらに基づく分枝限定法の実装及び数値実験結果について報告する．
    Sprache: Japanisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
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