Optimizing embedded systems, where the optimization of one depends on the state of another, is a formidable computational and algorithmic challenge, that is ubiquitous in real world systems. We study flow networks, where bilevel optimization is relevant to traffic planning, network control, and design, and where flows are governed by an optimization requirement subject to the network parameters. We employ message passing algorithms in flow networks with sparsely coupled structures to adapt network parameters that govern the network flows, in order to optimize a global objective. We demonstrate the effectiveness and efficiency of the approach on randomly generated graphs.
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B.L. and D.S. acknowledge support from the Leverhulme Trust (RPG-2018-092), European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 835913. B.L. acknowledges support from the startup funding from Harbin Institute of Technology, Shenzhen (Grant No. 20210134). D.S. acknowledges support from the EPSRC programme grant TRANSNET (EP/R035342/1). C.H.Y. is supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects No. EdUHK GRF 18304316, No. GRF 18301217, and No. GRF 18301119), the Dean's Research Fund of the Faculty of Liberal Arts and Social Sciences (Projects No. FLASS/DRF 04418, No. FLASS/ROP 04396, and No. FLASS/DRF 04624), and the Internal Research Grant (Projects No. RG67 2018-2019R R4015 and No. RG31 2020-2021R R4152), The Education University of Hong Kong, Hong Kong Special Administrative Region, China.