Semi-Decentralized Nash Equilibrium Seeking in Aggregative Games With Separable Coupling Constraints and Non-Differentiable Cost Functions

We study the Nash equilibrium seeking problem for noncooperative agents whose decision making process can be modeled as a generalized aggregative game. Specifically, we consider players with convex local cost functions, convex local constraints, and convex separable coupling constraints, and we extend the literature on generalized aggregative games by handling possibly non-differentiable cost functions. We recast the Nash equilibrium seeking problem as the problem to find a zero of a set-valued monotone operator and show that the variational Nash equilibria correspond to KKT solutions of the original game where the shared constraints have the same Lagrange multipliers for all the players. Finally, we design a semi-decentralized algorithm with global convergence guarantee for generalized Nash equilibrium seeking.

[1]  Sergio Grammatico.  Dynamic Control of Agents Playing Aggregative Games With Coupling Constraints , 2016, IEEE Transactions on Automatic Control.

[2]  Jorge Barrera,et al.  Dynamic Incentives for Congestion Control , 2015, IEEE Transactions on Automatic Control.

[3]  Francesca Parise,et al.  Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control , 2014, IEEE Transactions on Automatic Control.

[4]  Sergio Grammatico,et al.  On convexity and monotonicity in generalized aggregative games , 2017 .

[5]  Ankur A. Kulkarni,et al.  On the variational equilibrium as a refinement of the generalized Nash equilibrium , 2012, Autom..

[6]  Nikolai S. Kukushkin,et al.  Best response dynamics in finite games with additive aggregation , 2004, Games Econ. Behav..

[7]  Walid Saad,et al.  Game-Theoretic Methods for the Smart Grid: An Overview of Microgrid Systems, Demand-Side Management, and Smart Grid Communications , 2012, IEEE Signal Processing Magazine.

[8]  Munther A. Dahleh,et al.  Demand Response Using Linear Supply Function Bidding , 2015, IEEE Transactions on Smart Grid.

[9]  Sergio Grammatico,et al.  Exponentially convergent decentralized charging control for large populations of plug-in electric vehicles , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[10]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[11]  Francisco Facchinei,et al.  Modern Optimization Modelling Techniques , 2012, Advanced courses in mathematics : CRM Barcelona.

[12]  Andreas Fischer,et al.  On generalized Nash games and variational inequalities , 2007, Oper. Res. Lett..

[13]  Convex Optimization in Signal Processing and Communications , 2010 .

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Mauro Passacantando,et al.  Nash Equilibria, Variational Inequalities, and Dynamical Systems , 2002 .

[16]  RockaJellm.  MONOTONE OPERATORS ASSOCIATED WITH SADDLE . FUNCTIONS AND MINIMAX PROBLEMS R . 1 ' , 2022 .

[17]  Francesca Parise,et al.  Distributed computation of generalized Nash equilibria in quadratic aggregative games with affine coupling constraints , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[18]  Farzad Salehisadaghiani,et al.  Distributed Nash equilibrium seeking: A gossip-based algorithm , 2016, Autom..

[19]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[20]  Marc Teboulle,et al.  Lagrangian Duality and Related Multiplier Methods for Variational Inequality Problems , 1999, SIAM J. Optim..

[21]  Angelia Nedic,et al.  Distributed Algorithms for Aggregative Games on Graphs , 2016, Oper. Res..

[22]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[23]  W. Marsden.  I and J , 2012 .

[24]  Francisco Facchinei,et al.  Generalized Nash Equilibrium Problems , 2010, Ann. Oper. Res..

[25]  Prashant G. Mehta,et al.  Nash Equilibrium Problems With Scaled Congestion Costs and Shared Constraints , 2011, IEEE Transactions on Automatic Control.