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Differential variational inequality approach to dynamic games with shared constraints

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Abstract

The dynamic Nash equilibrium problem with shared constraints (NEPSC) involves a dynamic decision process with multiple players, where not only the players’ cost functionals but also their admissible control sets depend on the rivals’ decision variables through shared constraints. For a class of the dynamic NEPSC, we propose a differential variational inequality formulation. Using this formulation, we show the existence of solutions of the dynamic NEPSC, and develop a regularized smoothing method to find a solution of it. We prove that the regularized smoothing method converges to the least norm solution of the differential variational inequality, which is a solution of the dynamic NEPSC as the regularization parameter \(\lambda \) and smoothing parameter \(\mu \) go to zero with the order \(\mu =o(\lambda )\). Numerical examples are given to illustrate the existence and convergence results.

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Acknowledgments

We would like to thank Sven Leyffer, the anonymous associate editor and two referees for providing valuable comments and constructive suggestions, which help us to enrich the content and improve the presentation of the results in this paper. We are grateful to Jong-Shi Pang for his helpful comments on the optimal control problem with joint control and state constraints.

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Xiaojun Chen

  2. Department of Mathematics, Nanjing University, Nanjing, China

    Zhengyu Wang

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  1. Xiaojun Chen
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  2. Zhengyu Wang
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Correspondence to Xiaojun Chen.

Additional information

This work was supported partly by Hong Kong Research Grant Council PolyU5003/11p, the Fundamental Research Funds for the Central Universities (Grant No. 1113020301) and the National Natural Science Foundation of China (Grant No. 10871092).

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Chen, X., Wang, Z. Differential variational inequality approach to dynamic games with shared constraints. Math. Program. 146, 379–408 (2014). https://doi.org/10.1007/s10107-013-0689-1

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  • DOI: https://doi.org/10.1007/s10107-013-0689-1

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