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Article

Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target?

by
Godehard Sutmann
1,2
1
Jülich Supercomputing Centre (JSC), Forschungszentrum Jülich (FZJ), D-52425 Jülich, Germany
2
Interdisciplinary Center for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, D-44801 Bochum, Germany
Mathematics 2025, 13(20), 3269; https://doi.org/10.3390/math13203269 (registering DOI)
Submission received: 28 August 2025 / Revised: 30 September 2025 / Accepted: 7 October 2025 / Published: 13 October 2025
(This article belongs to the Special Issue Statistics for Stochastic Processes)

Abstract

Random walks are considered in a one-dimensional monotonously decreasing energy landscape. To reach the minimum within a region Ωϵ, a number of downhill steps have to be performed. A stochastic model is proposed which captures this random downhill walk and to make a prediction for the average number of steps, which are needed to hit the target. Explicit expressions in terms of a recurrence relation are derived for the density distribution of a downhill random walk as well as probability distribution functions to hit a target region Ωϵ within a given number of steps. For the case of stochastic optimisation, the number of rejected steps between two successive downhill steps is also derived, providing a measure for the average total number of trial steps. Analytical results are obtained for generalised random processes with underlying polynomial distribution functions. Finally the more general case of non-monotonously decreasing energy landscapes is considered for which results of the monotonous case are transferred by applying the technique of decreasing rearrangement. It is shown that the global stochastic optimisation can be fully described analytically, which is verified by numerical experiments for a number of different distribution and objective functions. Finally we discuss the transition to higher dimensional objective functions and discuss the change in computational complexity for the stochastic process.
Keywords: stochastic optimisation; random walks; random search stochastic optimisation; random walks; random search

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MDPI and ACS Style

Sutmann, G. Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics 2025, 13, 3269. https://doi.org/10.3390/math13203269

AMA Style

Sutmann G. Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics. 2025; 13(20):3269. https://doi.org/10.3390/math13203269

Chicago/Turabian Style

Sutmann, Godehard. 2025. "Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target?" Mathematics 13, no. 20: 3269. https://doi.org/10.3390/math13203269

APA Style

Sutmann, G. (2025). Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics, 13(20), 3269. https://doi.org/10.3390/math13203269

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