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May, 1983 Minimization Algorithms and Random Walk on the $d$-Cube
David Aldous
Ann. Probab. 11(2): 403-413 (May, 1983). DOI: 10.1214/aop/1176993605


Consider the number of steps needed by algorithms to locate the minimum of functions defined on the $d$-cube, where the functions are known to have no local minima except the global minimum. Regard this as a game: one player chooses a function, trying to make the number of steps needed large, while the other player chooses an algorithm, trying to make this number small. It is proved that the value of this game is approximately of order $2^{d/2}$ steps as $d \rightarrow \infty$. The key idea is that the hitting times of the random walk provide a random function for which no algorithm can locate the minimum within $2^{d(1/2 - \varepsilon)}$ steps.


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David Aldous. "Minimization Algorithms and Random Walk on the $d$-Cube." Ann. Probab. 11 (2) 403 - 413, May, 1983.


Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0513.60068
MathSciNet: MR690137
Digital Object Identifier: 10.1214/aop/1176993605

Primary: 60J15
Secondary: 68C25

Keywords: $d$-dimensional cube , computational complexity , Minimization algorithms , Random walk

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 2 • May, 1983
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