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A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

Author

Listed:
  • Jean-Baptiste Hiriart-Urruty

    (Université Paul Sabatier)

  • Jérôme Malick

    (CNRS)

Abstract

Engineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real numbers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, is spread all the way through. The “raison d’être” of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics.

Suggested Citation

  • Jean-Baptiste Hiriart-Urruty & Jérôme Malick, 2012. "A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 551-577, June.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:3:d:10.1007_s10957-011-9980-6
    DOI: 10.1007/s10957-011-9980-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2007. "Smoothing technique and its applications in semidefinite optimization," LIDAM Reprints CORE 1951, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
    3. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Alberto Seeger & David Sossa, 2015. "Complementarity problems with respect to Loewnerian cones," Journal of Global Optimization, Springer, vol. 62(2), pages 299-318, June.
    2. Baey, Charlotte & Cournède, Paul-Henry & Kuhn, Estelle, 2019. "Asymptotic distribution of likelihood ratio test statistics for variance components in nonlinear mixed effects models," Computational Statistics & Data Analysis, Elsevier, vol. 135(C), pages 107-122.
    3. Jean-Baptiste Hiriart-Urruty & Hai Le, 2013. "A variational approach of the rank function," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 207-240, July.
    4. Gerd Wachsmuth, 2015. "Mathematical Programs with Complementarity Constraints in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 480-507, August.

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