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arXiv · 1109.5241

Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms

Stephane Gaubert·William McEneaney·Zheng Qu
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Abstract

Max-plus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (max-plus "basis functions") taken from a prescribed dictionary. We study several variants of this approximation problem, which we show to be continuous versions of the facility location and $k$-center combinatorial optimization problems, in which the connection costs arise from a Bregman distance. We give theoretical error estimates, quantifying the number of basis functions needed to reach a prescribed accuracy. We derive from our approach a refinement of the curse of dimensionality free method introduced previously by McEneaney, with a higher accuracy for a comparable computational cost.

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Stephane Gaubert, William McEneaney, Zheng Qu. 2011-09-24. Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms. https://doi.org/10.1109/cdc.2011.6161386

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