Geodesic
Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian Manifolds
Journal of convex analysis, Tome 20 (2013) no. 2, pp. 395-438.

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We consider minimization problems with constraints. We show that if the set of constraints is a Riemannian manifold of non positive curvature and the objective function is lower semicontinuous and satisfies the Kurdyka-Lojasiewicz property, then the alternating proximal algorithm in Euclidean space is naturally extended to solve that class of problems. We prove that the sequence generated by our algorithm is well defined and converges to an inertial Nash equilibrium under mild assumptions about the objective function. As an application, we give a welcome result on the difficult problem of "learning how to play Nash" (convergence, convergence in finite time, speed of convergence, constraints in action spaces in the context of "alternating potential games" with inertia).
Classification : 65K10, 49J52, 49M27 , 90C26, 91B50, 91B06, 53B20
Mots-clés : Nash equilibrium, convergence, finite time, proximal algorithm, alternation, learning in games, inertia, Riemannian manifold, Kurdyka-Lojasiewicz property 
@article{JCA_2013_20_2_JCA_2013_20_2_a5,
     author = {J. X. Cruz Neto and P. R. Oliveira and P. A. Soares Jr and A. Soubeyran},
     title = {Learning how to {Play} {Nash,} {Potential} {Games} and {Alternating} {Minimization} {Method} for {Structured} {Nonconvex} {Problems} on {Riemannian} {Manifolds}},
     journal = {Journal of convex analysis},
     pages = {395--438},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2013},
     url = {http://geodesic.mathdoc.fr/item/JCA_2013_20_2_JCA_2013_20_2_a5/}
}
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J. X. Cruz Neto; P. R. Oliveira; P. A. Soares Jr; A. Soubeyran. Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian Manifolds. Journal of convex analysis, Tome 20 (2013) no. 2, pp. 395-438. http://geodesic.mathdoc.fr/item/JCA_2013_20_2_JCA_2013_20_2_a5/