Algorithm differentiation of implicit functions and optimal values 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.

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  1. Askham, Travis; Kutz, J. Nathan: Variable projection methods for an optimized dynamic mode decomposition (2018)
  2. van Leeuwen, Tristan; Maretzke, Simon; Batenburg, K. Joost: Automatic alignment for three-dimensional tomographic reconstruction (2018)
  3. Cocher, Emmanuel; Chauris, Hervé; Plessix, René-Édouard: Seismic iterative migration velocity analysis: two strategies to update the velocity model (2017)
  4. Gleixner, Ambros M.; Berthold, Timo; Müller, Benjamin; Weltge, Stefan: Three enhancements for optimization-based bound tightening (2017)
  5. Kasper Kristensen and Anders Nielsen and Casper Berg and Hans Skaug and Bradley Bell: TMB: Automatic Differentiation and Laplace Approximation (2016)
  6. Berthold, Timo: RENS. The optimal rounding (2014)
  7. Berthold, Timo; Gleixner, Ambros M.: Undercover: a primal MINLP heuristic exploring a largest sub-MIP (2014)
  8. Hogan, Robin J.: Fast reverse-mode automatic differentiation using expression templates in C++ (2014)
  9. Michael Braun: trustOptim: An R Package for Trust Region Optimization with Sparse Hessians (2014)
  10. Bell, Bradley M.; Flaxman, Abraham D.: A statistical model and estimation of disease rates as functions of age and time (2013)
  11. Gleixner, Ambros M.; Weltge, Stefan: Learning and propagating Lagrangian variable bounds for mixed-integer nonlinear programming (2013)
  12. Siehr, Jochen: Numerical optimization methods within a continuation strategy for the reduction of chemical combustion models (2013)
  13. Skanda, Dominik; Lebiedz, Dirk: A robust optimization approach to experimental design for model discrimination of dynamical systems (2013)
  14. Andersson, Joel; Åkesson, Johan; Diehl, Moritz: CasADi: A symbolic package for automatic differentiation and optimal control (2012)
  15. Aravkin, Aleksandr Y.; Van Leeuwen, Tristan: Estimating nuisance parameters in inverse problems (2012)
  16. Gleixner, Ambros M.; Held, Harald; Huang, Wei; Vigerske, Stefan: Towards globally optimal operation of water supply networks (2012)
  17. Lebiedz, Dirk; Siehr, Jochen; Unger, Jonas: A variational principle for computing slow invariant manifolds in dissipative dynamical systems (2011)
  18. Sielemann, M.; Schmitz, G.: A quantitative metric for robustness of nonlinear algebraic equation solvers (2011)
  19. Szynkiewicz, Wojciech; Błaszczyk, Jacek: Optimization-based approach to path planning for closed chain robot systems (2011)
  20. Fourer, Robert; Ma, Jun; Martin, Kipp: OSiL: An instance language for optimization (2010)

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