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EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES.

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Title: EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES.
Authors: HENRION, DIDIER1 henrion@laas.fr, NALDI, SIMONE2 caponord2007@gmail.com, EL DIN, MOHAB SAFEY3 mohab.safey@lip6.fr
Source: SIAM Journal on Optimization. 2016, Vol. 26 Issue 4, p2512-2539. 28p.
Subjects: Linear matrix inequalities, Algorithms, Polynomials, Quadratic programming, Topological degree
Abstract: Let A(x) = A0 + x1A1 +...+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0,A1,...,An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Optimization%22">SIAM Journal on Optimization</searchLink>. 2016, Vol. 26 Issue 4, p2512-2539. 28p.
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  Data: <searchLink fieldCode="DE" term="%22Linear+matrix+inequalities%22">Linear matrix inequalities</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+programming%22">Quadratic programming</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+degree%22">Topological degree</searchLink>
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  Data: Let A(x) = A0 + x1A1 +...+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0,A1,...,An of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of d. When m (resp., n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp., m). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Optimization is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/15M1036543
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      – Code: eng
        Text: English
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        PageCount: 28
        StartPage: 2512
    Subjects:
      – SubjectFull: Linear matrix inequalities
        Type: general
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Quadratic programming
        Type: general
      – SubjectFull: Topological degree
        Type: general
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      – TitleFull: EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES.
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              M: 12
              Text: 2016
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