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Solving rank-constrained semidefinite programs in exact arithmetic.

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Title: Solving rank-constrained semidefinite programs in exact arithmetic.
Authors: Naldi, Simone1 simone.naldi@tu-dortmund.de
Source: Journal of Symbolic Computation. Mar2018, Vol. 85, p206-223. 18p.
Subjects: Semidefinite programming, Linear complementarity problem, Polynomial time algorithms, Polynomial operators, Algorithmic randomness
Abstract: We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix, or the dimension of the affine section, is fixed, the algorithm is polynomial time. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We implement it in Maple and discuss practical experiments. [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: Solving rank-constrained semidefinite programs in exact arithmetic.
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Symbolic+Computation%22">Journal of Symbolic Computation</searchLink>. Mar2018, Vol. 85, p206-223. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Semidefinite+programming%22">Semidefinite programming</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+complementarity+problem%22">Linear complementarity problem</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomial+time+algorithms%22">Polynomial time algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomial+operators%22">Polynomial operators</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithmic+randomness%22">Algorithmic randomness</searchLink>
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  Data: We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix, or the dimension of the affine section, is fixed, the algorithm is polynomial time. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We implement it in Maple and discuss practical experiments. [ABSTRACT FROM AUTHOR]
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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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      – Type: doi
        Value: 10.1016/j.jsc.2017.07.009
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      – Code: eng
        Text: English
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        PageCount: 18
        StartPage: 206
    Subjects:
      – SubjectFull: Semidefinite programming
        Type: general
      – SubjectFull: Linear complementarity problem
        Type: general
      – SubjectFull: Polynomial time algorithms
        Type: general
      – SubjectFull: Polynomial operators
        Type: general
      – SubjectFull: Algorithmic randomness
        Type: general
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      – TitleFull: Solving rank-constrained semidefinite programs in exact arithmetic.
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              Text: Mar2018
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              Y: 2018
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              Value: 85
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            – TitleFull: Journal of Symbolic Computation
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