Deterministic Global Optimization over trained Kolmogorov Arnold Networks

To address the challenge of tractability for optimizing mathematical models in science and engineering, surrogate models are often employed. Recently, a new class of machine learning models named Kolmogorov Arnold Networks (KANs) have been proposed. It was reported that KANs can approximate a given input/output relationship with a high level of accuracy, requiring significantly fewer parameters than multilayer perceptrons. Hence, we aim to assess the suitability of deterministic global optimization of trained KANs by proposing their Mixed-Integer Nonlinear Programming (MINLP) formulation. We conduct extensive computational experiments for different KAN architectures. Additionally, we propose alternative convex hull reformulation, local support and redundant constraints for the formulation aimed at improving the effectiveness of the MINLP formulation of the KAN. KANs demonstrate high accuracy while requiring relatively modest computational effort to optimize them, particularly for cases with less than five inputs or outputs. For cases with higher inputs or outputs, carefully considering the KAN architecture during training may improve its effectiveness while optimizing over a trained KAN. Overall, we observe that KANs offer a promising alternative as surrogate models for deterministic global optimization.