Journal of Global Optimization

, Volume 57, Issue 1, pp 143–176 | Cite as

Improved relaxations for the parametric solutions of ODEs using differential inequalities

  • Joseph K. Scott
  • Paul I. Barton


A new method is described for computing nonlinear convex and concave relaxations of the solutions of parametric ordinary differential equations (ODEs). Such relaxations enable deterministic global optimization algorithms to be applied to problems with ODEs embedded, which arise in a wide variety of engineering applications. The proposed method computes relaxations as the solutions of an auxiliary system of ODEs, and a method for automatically constructing and numerically solving appropriate auxiliary ODEs is presented. This approach is similar to two existing methods, which are analyzed and shown to have undesirable properties that are avoided by the new method. Two numerical examples demonstrate that these improvements lead to significantly tighter relaxations than previous methods.


Convex relaxations Global optimization Optimal control 

Mathematics Subject Classification

34A40 65L05 49M20 49M37 90C26 


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Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Process Systems Engineering Laboratory, Department of Chemical EngineeringMITCambridgeUSA
  2. 2.Process Systems Engineering Laboratory, Department of Chemical EngineeringMITCambridgeUSA

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