Algorithmic 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.