Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations
Benamou, Jean-David; Carlier, Guillaume (2015), Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations, Journal of Optimization Theory and Applications, 167, 1, p. 1-26. http://dx.doi.org/10.1007/s10957-015-0725-9
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.inria.fr/hal-01073143
Journal nameJournal of Optimization Theory and Applications
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Abstract (EN)Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time-dependent continuity equation, which again can be formulated as a divergence constraint but in time and space. The variational class of mean field games, introduced by Lasry and Lions, may also be interpreted as a generalization of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well suited to treat such convex but non-smooth problems. They include in particular Monge historic optimal transport problem. A finite-element discretization and implementation of the method are used to provide numerical simulations and a convergence study.
Subjects / KeywordsOptimal transport; Mean field games; Monge problem; Degenerate elliptic PDEs; Augmented Lagrangian
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