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Cubic overestimation and secant updating for unconstrained optimization of C functions.

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Title: Cubic overestimation and secant updating for unconstrained optimization of C functions.
Authors: Griewank, Andreas1 (AUTHOR) griewank@math.hu-berlin.de, Fischer, Jonathan2 (AUTHOR), Bosse, Torsten1 (AUTHOR)
Source: Optimization Methods & Software. Aug2014, Vol. 29 Issue 5, p1075-1089. 15p.
Subjects: Estimation theory, Secant function, Mathematical optimization, Approximation theory, Mathematical models, Iterative methods (Mathematics)
Abstract: The discrepancy between an objective functionfand its local quadratic modelf(x)+∇f(x)⊤s+s⊤H(x)s/2 ≈f(x+s) at the current iteratexis estimated using a cubic termq|s|3/3. Potential steps are chosen such that they minimize (or at least significantly reduce) the overestimating function ∇f(x)⊤s+s⊤B s/2+q|s|3/3 withB≈H(x). This ensuresf(x+s)0 is too small. Either one or both quantities may be updated after unsuccessful and successful steps alike. For an algorithm employing both the symmetric rank one update and a shifted version of the BFGS formula we show that either∈f|∇f|=0 or sup |B|=∞, provided the HessianH(x) is Lipschitz on some neighbourhood of a bounded level set. Superlinear convergence is theoretically expected and numerically observed but not yet proven. [ABSTRACT FROM AUTHOR]
Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd 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: Cubic overestimation and secant updating for unconstrained optimization of C functions.
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  Data: <searchLink fieldCode="JN" term="%22Optimization+Methods+%26+Software%22">Optimization Methods & Software</searchLink>. Aug2014, Vol. 29 Issue 5, p1075-1089. 15p.
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  Data: <searchLink fieldCode="DE" term="%22Estimation+theory%22">Estimation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Secant+function%22">Secant function</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+theory%22">Approximation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink>
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  Data: The discrepancy between an objective functionfand its local quadratic modelf(x)+∇f(x)⊤s+s⊤H(x)s/2 ≈f(x+s) at the current iteratexis estimated using a cubic termq|s|3/3. Potential steps are chosen such that they minimize (or at least significantly reduce) the overestimating function ∇f(x)⊤s+s⊤B s/2+q|s|3/3 withB≈H(x). This ensuresf(x+s)<f(x) unless the approximating HessianB=B⊤differs significantly fromH(x) or the scalarq>0 is too small. Either one or both quantities may be updated after unsuccessful and successful steps alike. For an algorithm employing both the symmetric rank one update and a shifted version of the BFGS formula we show that either∈f|∇f|=0 or sup |B|=∞, provided the HessianH(x) is Lipschitz on some neighbourhood of a bounded level set. Superlinear convergence is theoretically expected and numerically observed but not yet proven. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd 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: 10.1080/10556788.2013.863308
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      – Code: eng
        Text: English
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        PageCount: 15
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    Subjects:
      – SubjectFull: Estimation theory
        Type: general
      – SubjectFull: Secant function
        Type: general
      – SubjectFull: Mathematical optimization
        Type: general
      – SubjectFull: Approximation theory
        Type: general
      – SubjectFull: Mathematical models
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      – SubjectFull: Iterative methods (Mathematics)
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      – TitleFull: Cubic overestimation and secant updating for unconstrained optimization of C functions.
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            NameFull: Griewank, Andreas
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            NameFull: Fischer, Jonathan
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            NameFull: Bosse, Torsten
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            – D: 01
              M: 08
              Text: Aug2014
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