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Norm-induced Cuts: Optimization with Lipschitzian Black-box Functions

Submission Status:under review
  • Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter. We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the “shape” of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example.

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Author:Adrian Göß, Alexander Martin, Sebastian Pokutta, Kartikey Sharma
Document Type:Preprint
Language:English
Date of Publication (online):2023/05/11
Release Date:2023/05/16
Tag:Black-box Optimization; Derivative-free Optimization; Global Optimization; Lipschitz Optimization
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Zuse-Institut Berlin (ZIB)
Technische Universität Nürnberg
Subprojects:A05
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International