Publication: Algorithmic Differentiation of Implicit Functions and Optimal Values
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Algorithmic Differentiation of Implicit Functions and Optimal Values

- incollection -
 

Author(s)
Bradley M. Bell , James V. Burke

Published in
Advances in Automatic Differentiation

Editor(s)
Christian H. Bischof, H. Martin Bücker, Paul D. Hovland, Uwe Naumann, J. Utke

Year
2008

Publisher
Springer

Abstract
In applied optimization, an understanding of the sensitivity of the optimal value to changes in structural parameters is often essential. Applications include parametric optimization, saddle point problems, Benders decompositions, and multilevel optimization. In this paper we adapt a known automatic differentiation (ad) technique for obtaining derivatives of implicitly defined functions for application to optimal value functions. The formulation we develop is well suited to the evaluation of first and second derivatives of optimal values. The result is a method that yields large savings in time and memory. The savings are demonstrated by a Benders decomposition example using both the ADOL-C and CppAD packages. Some of the source code for these comparisons is included to aid testing with other hardware and compilers, other ad packages, as well as future versions of ADOL-C and CppAD. The source code also serves as an aid in the implementation of the method for actual applications. In addition, it demonstrates how multiple C++ operator overloading ad packages can be used with the same source code. This provides motivation for the coding numerical routines where the floating point type is a C++ template parameter.

Cross-References
Bischof2008AiA

AD Tools
ADOL-C, CppAD

AD Theory and Techniques
Fixpoint

BibTeX
@INCOLLECTION{
         Bell2008ADo,
       title = "Algorithmic Differentiation of Implicit Functions and Optimal Values",
       doi = "10.1007/978-3-540-68942-3_7",
       author = "Bradley M. Bell and James V. Burke",
       abstract = "In applied optimization, an understanding of the sensitivity of the optimal value
         to changes in structural parameters is often essential. Applications include parametric
         optimization, saddle point problems, Benders decompositions, and multilevel optimization. In this
         paper we adapt a known automatic differentiation (AD) technique for obtaining derivatives of
         implicitly defined functions for application to optimal value functions. The formulation we develop
         is well suited to the evaluation of first and second derivatives of optimal values. The result is a
         method that yields large savings in time and memory. The savings are demonstrated by a Benders
         decomposition example using both the ADOL-C and CppAD packages. Some of the source code for these
         comparisons is included to aid testing with other hardware and compilers, other AD packages, as well
         as future versions of ADOL-C and CppAD. The source code also serves as an aid in the implementation
         of the method for actual applications. In addition, it demonstrates how multiple C++ operator
         overloading AD packages can be used with the same source code. This provides motivation for the
         coding numerical routines where the floating point type is a C++ template parameter.",
       crossref = "Bischof2008AiA",
       pages = "67--77",
       booktitle = "Advances in Automatic Differentiation",
       publisher = "Springer",
       editor = "Christian H. Bischof and H. Martin B{\"u}cker and Paul D. Hovland and Uwe
         Naumann and J. Utke",
       isbn = "978-3-540-68935-5",
       issn = "1439-7358",
       year = "2008",
       ad_tools = "ADOL-C, CppAD",
       ad_theotech = "Fixpoint"
}


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