Average performance of a class of adaptive algorithms for global optimization

Calvin, James M.

doi:10.1214/aoap/1034801250

Cited by 30 publications

(20 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Several optimization algorithms have been proposed that use the Brownian motion as a model for an unknown function to be minimized, including [10,12,21,3]. One of the ideas proposed in [10] is to evaluate the function next at the point where the function has the maximum conditional probability of having a value less than the minimum of the conditional mean, minus some positive amount (tending to zero).…”

Section: Introductionmentioning

confidence: 99%

Adaptive approximation of the minimum of Brownian motion

Calvin

Hefter

Herzwurm

2017

Journal of Complexity

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Adaptive approximation of the minimum of Brownian motion

Calvin

Hefter

Herzwurm

2017

Journal of Complexity

Self Cite

View full text Add to dashboard Cite

show abstract

“…uniform samples on (0, 1), independent of the Brownian Motion (X t ) t∈ [0,1] , [Calvin and Glynn, 1997] establish the weak limit of √ n ∆(T rnd n ). Finally, [Calvin, 1997] proposed a class of adaptive grids, meaning that the consecutive grid-points t k+1 are chosen based on ((t 1 , B t1 ), . .…”

Section: ∆(T ) = Supmentioning

confidence: 99%

Controlling the Time Discretization Bias for the Supremum of Brownian Motion

Bisewski

Crommelin

Mandjes

2018

ACM Trans. Model. Comput. Simul.

View full text Add to dashboard Cite

We consider the bias arising from time discretization when estimating the threshold crossing probability w(b) := P(sup t∈[0,1] Bt > b), with (Bt) t∈[0,1] a standard Brownian Motion. We prove that if the discretization is equidistant, then to reach a given target value of the relative bias, the number of grid points has to grow quadratically in b, as b grows. When considering non-equidistant discretizations (with threshold-dependent grid points), we can substantially improve on this: we show that for such grids the required number of grid points is independent of b, and in addition we point out how they can be used to construct a strongly efficient algorithm for the estimation of w(b). Finally, we show how to apply the resulting algorithm for a broad class of stochastic processes; it is empirically shown that the threshold-dependent grid significantly outperforms its equidistant counterpart. * Email: bisewski@cwi.nl 1 As b → ∞, both w T (b) and w(b) tend to 0, so that the absolute bias is not a meaningful accuracy measure.

show abstract

“…In the case of Brownian motion, it is shown in Calvin (1995) that if observations form a deterministic equispaced grid, then the error is about 82% as large as if the points are chosen at random uniformly over the unit interval. However, if new observations are to be added the uniformity of the deterministic grid will not hold at all times.…”

Section: Error Analysismentioning

confidence: 99%

Comparison of Monte Carlo and deterministic methods for non-adaptive optimization

Al-Mharmah

Calvin

1997

Proceedings of the 29th Conference on Winter Simulation - WSC '97

Self Cite

View full text Add to dashboard Cite

In this paper we compare the average performance of Monte Carlo methods for global optimization with non-adaptive deterministic alternatives. We analyze the behavior of the algorithms under the assumption of Wiener measure on the space of continuous functions on the unit interval. In this setting we show that the primary strength of the Monte Carlo methods (compositeness) is outweighed by the primary weakness (random gap size) when compared to efficient deterministic methods.

show abstract

Average performance of a class of adaptive algorithms for global optimization

Cited by 30 publications

References 11 publications

Adaptive approximation of the minimum of Brownian motion

Adaptive approximation of the minimum of Brownian motion

Controlling the Time Discretization Bias for the Supremum of Brownian Motion

Comparison of Monte Carlo and deterministic methods for non-adaptive optimization

Contact Info

Product

Resources

About