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Exact algorithms for semidefinite programs with degenerate feasible set.

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Title: Exact algorithms for semidefinite programs with degenerate feasible set.
Authors: Henrion, Didier1,2 (AUTHOR) henrion@laas.fr, Naldi, Simone3 (AUTHOR) simone.naldi@unilim.fr, Safey El Din, Mohab4 (AUTHOR) mohab.safey@lip6.fr
Source: Journal of Symbolic Computation. May2021, Vol. 104, p942-959. 18p.
Subjects: Semidefinite programming, Interior-point methods, Symmetric matrices, Algorithms, Polynomial time algorithms, Sum of squares
Abstract: Given symmetric matrices A 0 , A 1 , ... , A n of size m with rational entries, the set of real vectors x = (x 1 , ... , x n) such that the matrix A 0 + x 1 A 1 + ⋯ + x n A n has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Symbolic Computation is the property of Academic Press Inc. 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: Exact algorithms for semidefinite programs with degenerate feasible set.
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  Data: <searchLink fieldCode="AR" term="%22Henrion%2C+Didier%22">Henrion, Didier</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> henrion@laas.fr</i><br /><searchLink fieldCode="AR" term="%22Naldi%2C+Simone%22">Naldi, Simone</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> simone.naldi@unilim.fr</i><br /><searchLink fieldCode="AR" term="%22Safey+El+Din%2C+Mohab%22">Safey El Din, Mohab</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> mohab.safey@lip6.fr</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Symbolic+Computation%22">Journal of Symbolic Computation</searchLink>. May2021, Vol. 104, p942-959. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Semidefinite+programming%22">Semidefinite programming</searchLink><br /><searchLink fieldCode="DE" term="%22Interior-point+methods%22">Interior-point methods</searchLink><br /><searchLink fieldCode="DE" term="%22Symmetric+matrices%22">Symmetric matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomial+time+algorithms%22">Polynomial time algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Sum+of+squares%22">Sum of squares</searchLink>
– Name: Abstract
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  Data: Given symmetric matrices A 0 , A 1 , ... , A n of size m with rational entries, the set of real vectors x = (x 1 , ... , x n) such that the matrix A 0 + x 1 A 1 + ⋯ + x n A n has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of Journal of Symbolic Computation is the property of Academic Press Inc. 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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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.jsc.2020.11.001
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 18
        StartPage: 942
    Subjects:
      – SubjectFull: Semidefinite programming
        Type: general
      – SubjectFull: Interior-point methods
        Type: general
      – SubjectFull: Symmetric matrices
        Type: general
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Polynomial time algorithms
        Type: general
      – SubjectFull: Sum of squares
        Type: general
    Titles:
      – TitleFull: Exact algorithms for semidefinite programs with degenerate feasible set.
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            NameFull: Henrion, Didier
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            NameFull: Naldi, Simone
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            NameFull: Safey El Din, Mohab
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            – D: 01
              M: 05
              Text: May2021
              Type: published
              Y: 2021
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              Value: 104
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