Distributed Adaptive Resource Allocation: An Uncertain Saddle-Point Dynamics Viewpoint Dongdong Yue, Simone Baldi, Senior Member, IEEE, Jinde Cao, Fellow, IEEE, Qi Li, and Bart De Schutter, Fellow, IEEE Abstract—This paper addresses distributed adaptive optimal resource allocation problems over weight-balanced digraphs. By leveraging state-of-the-art adaptive coupling designs for multia- gent systems, two adaptive algorithms are proposed, namely a directed-spanning-tree-based algorithm and a node-based algo- rithm. The benefits of these algorithms are that they require nei- ther sufficiently small or unitary step sizes, nor global knowledge of Laplacian eigenvalues, which are widely required in the litera- ture. It is shown that both algorithms belong to a class of uncer- tain saddle-point dynamics, which can be tackled by repeatedly adopting the Peter-Paul inequality in the framework of Lya- punov theory. Thanks to this new viewpoint, global asymptotic convergence of both algorithms can be proven in a unified way. The effectiveness of the proposed algorithms is validated through numerical simulations and case studies in IEEE 30-bus and 118- bus power systems. Index Terms—Adaptive systems, directed graphs, resource alloca- tion, saddle-point dynamics.    I. Introduction T HE resource allocation problem, also known as the eco- nomic dispatch problem, has recently aroused multi-disci- plinary interest. Applications of resource allocation include various engineering fields such as cloud computing, sensor networks, and power systems. While early works studied opti- mal resource allocation based on a central node collecting and processing all data from every node in the network [1], this architecture is not effective in large-scale networks. There- fore, distributed resource allocation algorithms are highly desirable, i.e., to solve an allocation problem by making each node collect and process the data from only a few neighbor- ing nodes, according to the topology of the network. Different assumptions can be made on the graph describing the large-scale network: acyclic (tree) graph [2], undirected connected graph [3]–[12], strongly connected weight-bal- anced digraph [13]–[18], or weight-unbalanced digraph [19]–[21]. In most of these works, the algorithms used to solve the distributed resource allocation problem require uni- tary step sizes, or sufficiently small step sizes to implement local gradient descent, see e.g., [4]–[6], [17]–[20]. Meanwhile, many algorithms rely on homogeneous and static coupling gains, selected based on the global knowledge of Laplacian eigenvalues, e.g., [7], [10], [14]–[17], [21]. Such a strategy may lead to high-gain instability when the network is large and sparse (with a Laplacian eigenvalue being extremely close to the imaginary axis). Besides, for an effective distributed methodology, eliminating the global knowledge of the Lapla- cian matrix is crucial, which goes under the name of dis- tributed adaptive implementation. In fact, distributed adaptive algorithms incorporate adaptive (in place of static) coupling gains, which have the superiority of adapting to different network configurations. The reason is that these adaptive gains do not need to be selected based on global knowledge of Laplacian eigenvalues. Distributed adap- tive designs with adaptive coupling gains are available in the literature for consensus or tracking [22]–[26], containment or formation [27]–[29], and optimization [30], [31]. Distributed resource allocation solutions with adaptive cou- pling gains, to our best knowledge, are not available in the lit- erature, even for the simplest case of undirected graphs. The main reason for this gap lies in the following difficulty: In order to obtain an optimal resource allocation solution, the agents are supposed to seek a consensus over the Lagrangian multipliers based on a class of nested primal-dual dynamics [4]. This strategy brings the challenge of individual seeking of optimal allocation decisions and consensus seeking of the Lagrangian multipliers at the same time, without any knowl- edge of Laplacian eigenvalues. A possible approach to address this challenge is to solve the consensus optimization problem for the Lagrangian multipliers via distributed adaptive opti- mization of [30], [31]. Such an approach of focusing on the dual problem instead of the primal problem was indeed Manuscript received October 13, 2022; revised November 14, 2022; accepted December 20, 2022. This work was supported in part by the China Postdoctoral Science Foundation (BX2021064), the Fundamental Research Funds for the Central Universities (2242022R20030), the National Key R&D Program of China (2022YFE0198700), and the Natural Science Foundation of China (62150610499, 62073074, 61833005). Recommended by Associate Editor Hui Yu. (Corresponding authors: Dongdong Yue and Jinde Cao.) Citation: D. D. Yue, S. Baldi, J. D. Cao, Q. Li, and B. De Schutter, “Distributed adaptive resource allocation: An uncertain saddle-point dynamics viewpoint,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 12, pp. 2209–2221, Dec. 2023. D. D. Yue and S. Baldi are with the School of Mathematics, Southeast University, Nanjing 210096, China (e-mail: yued@seu.edu.cn; S.Baldi@ tudelft.nl). J. D. Cao is with the School of Mathematics, Southeast University, Nanjing 210096, China, and also with the Yonsei Frontier Laboratory, Yonsei University, Seoul 03722, South Korea (e-mail: jdcao@seu.edu.cn). Q. Li is with the School of Automation, Southeast University, Nanjing 210096, China (e-mail: liqikj@hotmail.com). B. De Schutter is with the Delft Center for Systems and Control, Delft University of Technology, Delft 2628 CD, The Netherlands (e-mail: B.De Schutter@tudelft.nl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2023.123402 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 10, NO. 12, DECEMBER 2023 2209