(194d) Reachability Bounds for Global and Robust Optimization of Parametric ODEs Via Implicit Methods
We present a novel approach to calculating relaxations of the set of solutions of ordinary differential equations subject to parametric uncertainty and uncertain initial values. This approach allows for the relaxation of pODE solution sets via parametric linear multistep methods, constructed in a sequential-block fashion. Each block is relaxed via the implicit method developed in . We show that this algorithm exhibits partition convergence. As a consequence of this result, our method exhibits the favorable properties inherent to generalized McCormick relaxations: tightness of the relaxations, enhanced convergence speed , and a propensity to avoid clustering about global minima .
The relaxations developed herein have been implemented as an extension to the EAGO software package in Julia [13, 14]. A performance assessment was made using a series of examples common in the extant literature. In multiple cases, these new techniques allow for an improved solution time when used within a deterministic global optimization context. As a further result, we show that this new approach can be readily applied to semi-infinite programs using the approach presented in . Finally, we solve select worst-case design problems with pODE constraints.
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