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Proving inequalities and solving global optimization problems via simplified CAD projection.

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Title: Proving inequalities and solving global optimization problems via simplified CAD projection.
Authors: Han, Jingjun1 hanjingjunfdfz@gmail.com, Jin, Zhi1,2 j2i5nzhi@sina.com, Xia, Bican1 xbc@math.pku.edu.cn
Source: Journal of Symbolic Computation. Jan2016, Vol. 72, p206-230. 25p.
Subjects: Geometrical drawing, Global optimization, Mathematical optimization, Computer-aided design, Algorithms
Abstract: Let x n = ( x 1 , … , x n ) and f ∈ R [ x n , k ] . The problem of finding all k 0 such that f ( x n , k 0 ) ≥ 0 on R n is considered in this paper, which obviously takes as a special case the problem of computing the global infimum or proving the semi-definiteness of a polynomial. For solving the problems, we propose a simplified Brown–McCallum's CAD projection operator, Nproj , of which the projection scale is always no larger than that of Brown–McCallum's. For many problems, the projection scale is much smaller than that of Brown–McCallum's. As a result, the lifting phase is also simplified. Some new algorithms based on Nproj for solving those problems are designed and proved to be correct. Comparison to some existing tools on some examples is reported to illustrate the effectiveness of our new algorithms. [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.)
Database: Engineering Source
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DbLabel: Engineering Source
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PubType: Academic Journal
PubTypeId: academicJournal
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  Data: Proving inequalities and solving global optimization problems via simplified CAD projection.
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  Data: <searchLink fieldCode="AR" term="%22Han%2C+Jingjun%22">Han, Jingjun</searchLink><relatesTo>1</relatesTo><i> hanjingjunfdfz@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Jin%2C+Zhi%22">Jin, Zhi</searchLink><relatesTo>1,2</relatesTo><i> j2i5nzhi@sina.com</i><br /><searchLink fieldCode="AR" term="%22Xia%2C+Bican%22">Xia, Bican</searchLink><relatesTo>1</relatesTo><i> xbc@math.pku.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Symbolic+Computation%22">Journal of Symbolic Computation</searchLink>. Jan2016, Vol. 72, p206-230. 25p.
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  Data: <searchLink fieldCode="DE" term="%22Geometrical+drawing%22">Geometrical drawing</searchLink><br /><searchLink fieldCode="DE" term="%22Global+optimization%22">Global optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Computer-aided+design%22">Computer-aided design</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink>
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  Label: Abstract
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  Data: Let x n = ( x 1 , … , x n ) and f ∈ R [ x n , k ] . The problem of finding all k 0 such that f ( x n , k 0 ) ≥ 0 on R n is considered in this paper, which obviously takes as a special case the problem of computing the global infimum or proving the semi-definiteness of a polynomial. For solving the problems, we propose a simplified Brown–McCallum's CAD projection operator, Nproj , of which the projection scale is always no larger than that of Brown–McCallum's. For many problems, the projection scale is much smaller than that of Brown–McCallum's. As a result, the lifting phase is also simplified. Some new algorithms based on Nproj for solving those problems are designed and proved to be correct. Comparison to some existing tools on some examples is reported to illustrate the effectiveness of our new algorithms. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  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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        Value: 10.1016/j.jsc.2015.02.007
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      – Code: eng
        Text: English
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        PageCount: 25
        StartPage: 206
    Subjects:
      – SubjectFull: Geometrical drawing
        Type: general
      – SubjectFull: Global optimization
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      – SubjectFull: Mathematical optimization
        Type: general
      – SubjectFull: Computer-aided design
        Type: general
      – SubjectFull: Algorithms
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      – TitleFull: Proving inequalities and solving global optimization problems via simplified CAD projection.
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            NameFull: Han, Jingjun
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            NameFull: Jin, Zhi
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              M: 01
              Text: Jan2016
              Type: published
              Y: 2016
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