Nonlinear convex and concave relaxations for the solutions of parametric ODEs

TitleNonlinear convex and concave relaxations for the solutions of parametric ODEs
Publication TypeJournal Article
Year of Publication2013
AuthorsScott JK, Chachuat B, Barton PI
JournalOptimal Control Applications and Methods
Volume34
Pagination145-163
ISSN1099-1514
Keywordsdynamic optimization, global optimization, optimal control, ordinary differential equations
Abstract

Convex and concave relaxations for the parametric solutions of ordinary differential equations (ODEs) are central to deterministic global optimization methods for nonconvex dynamic optimization and open-loop optimal control problems with control parametrization. Given a general system of ODEs with parameter dependence in the initial conditions and right-hand sides, this work derives sufficient conditions under which an auxiliary system of ODEs describes convex and concave relaxations of the parametric solutions, pointwise in the independent variable. Convergence results for these relaxations are also established. A fully automatable procedure for constructing an appropriate auxiliary system has been developed previously by the authors. Thus, the developments here lead to an efficient, automatic method for computing convex and concave relaxations for the parametric solutions of a very general class of nonlinear ODEs. The proposed method is presented in detail for a simple example problem.

URLhttp://dx.doi.org/10.1002/oca.2014
DOI10.1002/oca.2014