Exponential convergence for a convexifying equation and a non-autonomous gradient ow for global minimization

Guillaume Carlier () and Alfred Galichon ()
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Guillaume Carlier: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Alfred Galichon: ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique

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Abstract: We consider an evolution equation similar to that introduced by Vese in [10] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope.

Date: 2012-07
Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-01024585v1
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Published in Control, Optimisation and Calculus of Variations, 2012, 18 (3), pp.611-620

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