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@ARTICLE{Sutmann:1047010,
      author       = {Sutmann, Godehard},
      title        = {{S}tatistics of {G}lobal {S}tochastic {O}ptimization: how
                      many steps to hit the target?},
      journal      = {Mathematics},
      volume       = {13},
      number       = {20},
      issn         = {2227-7390},
      address      = {Basel},
      publisher    = {MDPI},
      reportid     = {FZJ-2025-04066},
      pages        = {3269},
      year         = {2025},
      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.},
      cin          = {JSC},
      ddc          = {510},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {5111 - Domain-Specific Simulation $\&$ Data Life Cycle Labs
                      (SDLs) and Research Groups (POF4-511)},
      pid          = {G:(DE-HGF)POF4-5111},
      typ          = {PUB:(DE-HGF)16},
      doi          = {10.3390/math13203269},
      url          = {https://juser.fz-juelich.de/record/1047010},
}