New inequalities for comparing ordinary differential equations arising in global dynamic optimization

Deterministic methods for global optimization typically rely on convex relaxations to infer crucial bounding information. In global dynamic optimization, comparing convex relaxation approaches can be challenging due to limitations in established differential inequality theory. In this article, we provide new comparison results for the Carathéodory solutions of related ordinary differential equations (ODEs). Our results are applicable to certain ODEs whose right-hand side functions need not be quasimonotonic or continuous with respect to state variables; they require only a weakened variant of the standard Lipschitz continuity assumption, along with mild differential inequality requirements motivated by interval analysis. Using our new comparison results, we reveal an intuitive monotonicity result for global dynamic optimization that was previously unknown: when constructing convex relaxations of parametric ODE solutions within a relaxation framework by Scott and Barton (J. Glob. Optim. 57:143–176, 2013), supplying tighter relaxations of the ODE right-hand sides will always translate into relaxations of the ODE solutions that are at least as tight.