Optimization-based convex relaxations for nonconvex parametric systems of ordinary differential equations

Abstract

Novel convex and concave relaxations are proposed for the solutions of parametric ordinary differential equations (ODEs), to aid in furnishing bounding information for deterministic global dynamic optimization methods. These relaxations are constructed as the solutions of auxiliary ODE systems with embedded convex optimization problems, whose objective functions employ convex and concave relaxations of the original ODE right-hand side. Unlike established relaxation methods, any valid convex and concave relaxations for the original right-hand side are permitted in the approach, including the McCormick relaxations and the $$\alpha $$ BB relaxations. In general, tighter such relaxations will necessarily translate into tighter relaxations for the ODE solutions, thus providing tighter bounding information for an overarching global dynamic optimization method. Notably, if McCormick relaxations are employed in the new approach, then the obtained relaxations are guaranteed to be at least as tight as state-of-the-art ODE relaxations based on generalized McCormick relaxations. The new relaxations converge rapidly to the original system as the considered parametric subdomain shrinks. Several examples are presented for comparison with established ODE relaxations, based on a proof-of-concept implementation in MATLAB. In a global optimization case study, the new ODE relaxations are shown to lead to fewer branch-and-bound global optimization iterations than state-of-the-art relaxations.