This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Matérn Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Matérn prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.

翻译：本文研究以度量图建模的网络上的优化问题。受目标函数评估代价高昂或仅能作为黑箱获取的应用场景启发，我们开发了贝叶斯优化算法，通过顺序更新目标函数的高斯过程代理模型来指导查询点的获取。为确保代理模型适应网络几何特性，我们采用通过度量图上随机偏微分方程定义的Whittle-Matérn高斯过程先验模型。除了建立优化足够光滑目标函数的遗憾界外，我们还分析了目标函数光滑性未知且Whittle-Matérn先验采用有限元表示的实际情形。数值结果表明，该算法在合成度量图上优化基准目标函数以及通过电信网络的最大后验估计进行贝叶斯反演时均具有有效性。