Staged construction of adjoints for constrained optimization of integro-differential boundary-value problems

The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirrors the natural decomposition of the original constraints. Second, it details a library of realizations of the zero functions, monitor functions, and contributions to the adjoint equations associated with each stage of problem construction. Third, it illustrates the application of this framework to multisegment trajectory problems, for example, in the analysis of spacecraft optimal control problems. The proposed construction paradigm generalizes a formulation for the construction of integro-differential-algebraic continuation problems in the software package COCO to also include automatically generated adjoint terms. The implementation enables arbitrarily many levels of nested construction with larger problems assembled from smaller ones, and multiple instances of a problem class combined inside a composite problem with the help of suitably defined coupling conditions. The application of this paradigm to problems of constrained optimization relies on the successive continuation paradigm introduced by Kernévez and Doedel [Optimization in bifurcation problems using a continuation method, in Bifurcation: Analysis, Algorithms, Applications, Birkhäuser Verlag, Basel, 1987, pp. 153-160], in which solutions to the necessary conditions for locally optimal solutions are found at the end of a sequence of easily initialized separate stages of continuation. The numerical examples illustrate the paradigm of construction and the efficacy of the continuation approach in locating stationary solutions to high-dimensional problems.