Skip to main content

Global Random Search in High Dimensions

  • 444 Accesses

Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)

Abstract

It is not the aim of this chapter to cover the whole subject of the global random search (GRS). It only contains some potentially important notes on algorithms of GRS in continuous problems, mostly keeping in mind the use of such algorithms in reasonably large dimensions. These notes are based on the 40-year experience of the author of this chapter and reflect his subjective opinions, some of which other specialists in the field do not necessarily share. The chapter discusses new results as well as rather old ones thoroughly explained in the monographs [49, 51, 53].

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-64712-4_3
  • Chapter length: 30 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-64712-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)
Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
Fig. 3.5
Fig. 3.6
Fig. 3.7
Fig. 3.8
Fig. 3.9

References

  1. A. Auger, B. Doerr (eds.), Theory of Randomized Search Heuristics: Foundations and Recent Developments (World Scientific, Singapore, 2011)

    MATH  Google Scholar 

  2. R. Battiti, M. Brunato, The lion way: machine learning plus intelligent optimization. LIONlab, University of Trento, Italy, 94 (2014)

    Google Scholar 

  3. Z. Beheshti, S.M.H. Shamsuddin, A review of population-based meta-heuristic algorithms. Int. J. Adv. Soft Comput. Appl. 5(1), 1–35 (2013)

    Google Scholar 

  4. J. Branke, Evolutionary Optimization in Dynamic Environments (Springer, Berlin, 2012)

    MATH  Google Scholar 

  5. P. Cooke, Optimal linear estimation of bounds of random variables. Biometrika 67, 257–258 (1980)

    MathSciNet  MATH  CrossRef  Google Scholar 

  6. L. De Haan, L. Peng, Comparison of tail index estimators. Statistica Neerlandica 52(1), 60–70 (1998)

    MathSciNet  MATH  CrossRef  Google Scholar 

  7. K. Deb, An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2–4), 311–338 (2000)

    MATH  CrossRef  Google Scholar 

  8. H. Dette, A. Pepelyshev, A. Zhigljavsky, Optimal design for linear models with correlated observations. Ann. Stat. 41(1), 143–176 (2013)

    MathSciNet  MATH  CrossRef  Google Scholar 

  9. H. Dette, A. Pepelyshev, A. Zhigljavsky, Optimal designs in regression with correlated errors. Ann. Stat. 44(1), 113 (2016)

    Google Scholar 

  10. K.-L. Du, M. Swamy, Search and Optimization by Metaheuristics (Birkhauser, Basel, 2016)

    MATH  CrossRef  Google Scholar 

  11. N. Dunford, J.T. Schwartz, Linear Operators. Part I (Wiley, New York, 1988)

    Google Scholar 

  12. M. Gendreau, J.-Y. Potvin, others (eds.), Handbook of Metaheuristics, Vol. 2 (Springer, New York, 2010)

    Google Scholar 

  13. J. Gillard, K. Usevich, Structured low-rank matrix completion for forecasting in time series analysis. Int. J. Forecast. 34(4), 582–597 (2018)

    CrossRef  Google Scholar 

  14. J. Gillard, A. Zhigljavsky, Optimization challenges in the structured low rank approximation problem. J. Global Optim. 57(3), 733–751 (2013)

    MathSciNet  MATH  CrossRef  Google Scholar 

  15. J. Gillard, A. Zhigljavsky, Stochastic algorithms for solving structured low-rank matrix approximation problems. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 70–88 (2015)

    MathSciNet  MATH  CrossRef  Google Scholar 

  16. J. Gillard, A. Zhigljavsky, Weighted norms in subspace-based methods for time series analysis. Numer. Linear Algebra Appl. 23(5), 947–967 (2016)

    MathSciNet  MATH  CrossRef  Google Scholar 

  17. J. Gillard, A. Zhigljavsky, Optimal directional statistic for general regression. Stat. Probab. Lett. 143, 74–80 (2018)

    MathSciNet  MATH  CrossRef  Google Scholar 

  18. J. Gillard, A. Zhigljavsky, Optimal estimation of direction in regression models with large number of parameters. Appl. Math. Comput. 318, 281–289 (2018)

    MATH  Google Scholar 

  19. F.W. Glover, G.A. Kochenberger (eds.), Handbook of Metaheuristics (Springer, New York, 2006)

    MATH  Google Scholar 

  20. D.E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning (Addison-Wesley, Reading, 1989)

    MATH  Google Scholar 

  21. E.M.T. Hendrix, O. Klepper, On uniform covering, adaptive random search and raspberries. J. Global Optim. 18(2), 143–163 (2000)

    MathSciNet  MATH  CrossRef  Google Scholar 

  22. E.M.T. Hendrix, B.G.-Tóth, Introduction to Nonlinear and Global Optimization, Vol. 37 (Springer, New York, 2010)

    Google Scholar 

  23. J. Hooker, Testing heuristics: we have it all wrong. J. Heuristics 1, 33–42 (1995)

    MATH  CrossRef  Google Scholar 

  24. N. Ketkar, Stochastic gradient descent. In: Deep Learning with Python, pp 113–132 (Springer, New York, 2017)

    Google Scholar 

  25. M.A. Krasnoselskij, J.A. Lifshits, A.V. Sobolev, Positive Linear Systems (Heldermann Verlag, Berlin, 1989)

    Google Scholar 

  26. P. Kulczycki, S. Lukasik, An algorithm for reducing the dimension and size of a sample for data exploration procedures. Int. J. Appl. Math. Comput. Sci. 24(1), 133–149 (2014)

    MathSciNet  MATH  CrossRef  Google Scholar 

  27. V.B. Nevzorov, Records: Mathematical Theory (American Mathematical Society, Providence, 2001)

    Google Scholar 

  28. J. Noonan, A. Zhigljavsky, Appriximation of the covering radius in high dimensions (2021, in preparation)

    Google Scholar 

  29. P. Pardalos, A. Zhigljavsky, J. Žilinskas, Advances in Stochastic and Deterministic Global Optimization (Springer, Switzerland, 2016)

    MATH  CrossRef  Google Scholar 

  30. N.R. Patel, R.L. Smith, Z.B. Zabinsky, Pure adaptive search in Monte Carlo optimization. Math. Program. 43(1–3), 317–328 (1989)

    MathSciNet  MATH  CrossRef  Google Scholar 

  31. M. Pelikan, Hierarchical Bayesian Optimization Algorithm (Springer, Berlin, Heidelberg, 2005)

    MATH  CrossRef  Google Scholar 

  32. A. Pepelyshev, A. Zhigljavsky, A. Žilinskas, Performance of global random search algorithms for large dimensions. J. Global Optim. 71(1), 57–71 (2018)

    MathSciNet  MATH  CrossRef  Google Scholar 

  33. J. Pintér, Convergence properties of stochastic optimization procedures. Optimization 15(3), 405–427 (1984)

    MathSciNet  MATH  Google Scholar 

  34. J. Pinter, Global Optimization in Action (Kluwer Academic Publisher, Dordrecht, 1996)

    MATH  CrossRef  Google Scholar 

  35. L. Pronzato, A. Zhigljavsky, Algorithmic construction of optimal designs on compact sets for concave and differentiable criteria. J. Stat. Plann. Inference 154, 141–155 (2014)

    MathSciNet  MATH  CrossRef  Google Scholar 

  36. C.R. Reeves, J.E. Rowe, Genetic Algorithms: Principles and Perspectives (Kluwer, Boston, 2003)

    MATH  Google Scholar 

  37. C. Ribeiro, P. Hansen (eds.), Essays and Surveys in Metaheuristics (Springer, New York, 2012)

    Google Scholar 

  38. A.H.G. Rinnooy Kan, G.T. Timmer, Stochastic global optimization methods. Part I: clustering methods. Math. Program. 39(1), 27–56 (1987)

    MathSciNet  MATH  Google Scholar 

  39. S.K. Sahu, A. Zhigljavsky, Self-regenerative Markov chain Monte Carlo with adaptation. Bernoulli 9(3), 395–422 (2003)

    MathSciNet  MATH  CrossRef  Google Scholar 

  40. D. Simon, Evolutionary Optimization Algorithms (Wiley, Chichester, 2013)

    Google Scholar 

  41. F. Solis, R. Wets, Minimization by random search techniques. Math. Oper. Res. 6(1), 19–30 (1981)

    MathSciNet  MATH  CrossRef  Google Scholar 

  42. D. Tarłowski, On the convergence rate issues of general Markov search for global minimum. J. Global Optim. 69(4), 869–888 (2017)

    MathSciNet  MATH  CrossRef  Google Scholar 

  43. A.S. Tikhomirov, On the convergence rate of the simulated annealing algorithm. Comput. Math. Math. Phys. 50(1), 19–31 (2010)

    MathSciNet  MATH  CrossRef  Google Scholar 

  44. A. Tikhomirov, T. Stojunina, V. Nekrutkin, Monotonous random search on a torus: integral upper bounds for the complexity. J. Stat. Plann. Inference 137(12), 4031–4047 (2007)

    MathSciNet  MATH  CrossRef  Google Scholar 

  45. A. Törn, A. Žilinskas, Global Optimization (Springer, Berlin, 1989)

    MATH  CrossRef  Google Scholar 

  46. W. Tu, W. Mayne, Studies of multi-start clustering for global optimization. Int. J. Numer. Meth. Eng. 53, 2239—2252 (2002)

    MathSciNet  MATH  CrossRef  Google Scholar 

  47. P. Van Laarhoven, E. Aarts, Simulated Annealing: Theory and Applications (Kluwer, Dordrecht, 1987)

    MATH  CrossRef  Google Scholar 

  48. Z.B. Zabinsky, Stochastic Adaptive Search for Global Optimization (Kluwer, Boston, 2003)

    MATH  CrossRef  Google Scholar 

  49. A. Zhigljavsky, Mathematical Theory of Global Random Search (Leningrad University Press, Leningrad, 1985). in Russian

    Google Scholar 

  50. A. Zhigljavsky, Branch and probability bound methods for global optimization. Informatica 1(1), 125–140 (1990)

    MathSciNet  MATH  Google Scholar 

  51. A. Zhigljavsky, Theory of Global Random Search (Kluwer, Dordrecht, 1991)

    CrossRef  Google Scholar 

  52. A. Zhigljavsky, E. Hamilton, Stopping rules in k-adaptive global random search algorithms. J. Global Optim. 48(1), 87–97 (2010)

    MathSciNet  MATH  CrossRef  Google Scholar 

  53. A. Zhigljavsky, A. Žilinskas, Stochastic Global Optimization (Springer, New York, 2008)

    MATH  Google Scholar 

  54. R. Zieliński, A statistical estimate of the structure of multi-extremal problems. Math. Program. 21, 348–356 (1981)

    MathSciNet  MATH  CrossRef  Google Scholar 

  55. A. Žilinskas, A. Zhigljavsky, Branch and probability bound methods in multi-objective optimization. Optim. Lett. 10(2), 1–13 (2016)

    MathSciNet  MATH  CrossRef  Google Scholar 

  56. A. Žilinskas, J. Gillard, M. Scammell, A. Zhigljavsky, Multistart with early termination of descents. J. Global Optim. 1–16 (2019). https://doi.org/10.1007/s10898-019-00814-w

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 The Author(s)

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Zhigljavsky, A., Žilinskas, A. (2021). Global Random Search in High Dimensions. In: Bayesian and High-Dimensional Global Optimization. SpringerBriefs in Optimization. Springer, Cham. https://doi.org/10.1007/978-3-030-64712-4_3

Download citation