**Project Heads**

Pavel Dvurechenskii, Caroline Geiersbach, Michael Hintermüller, Aswin Kannan

**Project Members**

Felix Sauer

**Project Duration**

01.04.2023 − 31.03.2026

**Located at**

WIAS

**Additional Information**

On the mathematical side, we study physics-based multiagent problems that take a form of stochastic generalized Nash equilibrium problems (SGNEPs) in infinite dimensions. In lieu of natural gas markets, literature has focused on parts or stylized versions of these problems, but have not been complete. Works related to game theoretic electricity markets on the other hand have not focused much on PDE type constraints or infinite dimensions. Study of such comprehensive and coupled systems is the major objective of this research.

The project is expected to advance the development of distributed models and methods in the context of multi-modal energy systems. From an application perspective, the goal is to model the intraday behavior of a distributed electricity market that is coupled to a hydrogen gas network. The mathematical goals include the rigorous analysis of such systems, including incorporating uncertainties and showing existence of equilibria for the distributed system in the function space setting. Additionally, novel mesh-independent distributed methods will be developed to compute equilibria.

**Related Publications
**

(1) A. Rogozin, A. Beznosikov, D. Dvinskikh, D. Kovalev, P. Dvurechensky, and A. Gasnikov, Decentralized saddle point problems via non-Euclidean mirror prox, Optimization Methods and Software, DOI: 10.1080/10556788.2023.2280062, 2024.

(2) P. Dvurechensky and J.-J. Zhu. Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research. PMLR, doi: 10.20347/WIAS.PREPRINT.3032, 2024.

(3) C. Geiersbach, R. Henrion, and P. Pérez-Aros, Numerical solution of an optimal control problem with probabilistic and almost sure state constraints, WIAS Preprint 3062, 2023.

(4) C. Geiersbach and R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Accepted in Mathematics of Operations Research, 2024.

(5) C. Geiersbach and M. Hintermüller, Optimality conditions and Moreau–Yosida regularization for almost sure state constraints, ESAIM: Control, Optimisation and Calculus of Variations, Volume 28, 2022.

(6) V. Grimm, M. Hintermüller, O. Huber, L. Schewe, M. Schmidt, and G. Zöttl, A PDE-constrained generalized Nash equilibrium approach for modeling gas markets with transport, Preprint, TRR 154, 2021.

(7) P. Dvurechensky, D. Dvinskikh, A. Gasnikov, C. Uribe, and A. Nedich, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, In NeurIPS 2018, pages 10783–10793, 2018.

(8) A. Kannan, U. Shanbhag, and H. Kim, Addressing supply-side risk in uncertain power markets: Stochastic Nash models, scalable algorithms and error analysis. Optimization Methods and Software, 28(5):1095–1138, 2013.

**Related Pictures
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