Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations

Jean-David Benamou and Guillaume Carlier ()
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Jean-David Benamou: MOKAPLAN, INRIA
Guillaume Carlier: U. Paris Dauphine Université Paris Dauphine

Journal of Optimization Theory and Applications, 2015, vol. 167, issue 1, No 1, 26 pages

Abstract: Abstract 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.

Keywords: Augmented Lagrangian; Optimal transport; Monge problem; Mean field games; Degenerate elliptic PDEs; 49A50; 49Q20; 65M60; 60K30 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-015-0725-9

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