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The robust constant and its applications in random global search for unconstrained global optimization

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Abstract

Robust analysis is important for designing and analyzing algorithms for global optimization. In this paper, we introduce a new concept, robust constant, to quantitatively characterize the robustness of measurable sets and functions. The new concept is consistent to the theoretical robustness presented in literatures. This paper shows that, from the respects of convergence theory and numerical computational cost, robust constant is valuable significantly for analyzing random global search methods for unconstrained global optimization.

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Notes

  1. The robustness of a function defined by Zheng [32] will be introduced in the next section.

References

  1. Appel, M.J., Labarre, R., Radulovic, D.: On accelerated random search. SIAM J. Optim. 14, 708–731 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baritompa, W., Zhang, B.P., Wood, G.R., Zabinsky, Z.B.: Towards pure adaptive search-a general framework and a one-dimensional realisation. J. Glob. Optim. 7, 93–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. De Boer, P.T., Kroese, D.P., Mannor, S., Rubinstein, R.Y.: A tutorial on the cross-entropy method. Ann. Oper. Res. 134, 19–67 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brooks, S.H.: A discussion of random methods for seeking maxima. Oper. Res. 6, 244–251 (1958)

    Article  MathSciNet  Google Scholar 

  5. Bulger, D.W., Wood, G.R.: Hesitant adaptive search for global optimisation. Math. Progr. 81, 89–102 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Bulger, D., Baritompa, W., Wood, G.R.: Implementing pure adaptive search with Grover’s quantum algorithm. J. Optim. Theory Appl. 116(3), 517–529 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chew, S., Zheng, Q.: Integral Global Optimization (Theory, Implementation and Applications). Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  8. Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, J., Fu, M.C., Marcus, S.I.: A model reference adaptive search method for global optimization. Oper. Res. 55(3), 549–568 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kabirian, A.: Hybrid probabilistic search methods for simulation optimization. J. Ind. Syst. Eng. 2(4), 259–270 (2009)

    Google Scholar 

  11. Kroese, D.P., Porotsky, S., Rubinstein, R.Y.: The cross-entropy method for continuous multi-extremal optimization. Methodol. Comput. Appl. Probab. 8, 383–407 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pardalos, P.M., Romeijn, H.E., Tuy, H.: Recent developments and trends in global optimization. J. Comput. Appl. Math. 124, 209–228 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Patel, N.R., Smith, R.L., Zabinsky, Z.B.: Pure adaptive search in Monte-Carlo optimization. Math. Progr. 43, 317–328 (1988)

    Article  MathSciNet  Google Scholar 

  14. Peng, Z., Wu, D., Zheng, Q.: A level-value estimation method and stochastic implementation for global optimization. J. Optim. Theory Appl. 156(2), 493–523 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peng, Z., Shen, Y., Wu, D.H.: A modified integral global optimization method and its asymptotic convergence. Acta Math. Appl. Sin (Engl. Ser.) 25(2), 283–290 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pinter, J.: Convergence qualification of adaptive partition algorithms in global optimization. Math. Progr. 56, 343–360 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Price, C.J., Reale, M., Robertson, B.L.: A cover partitioning method for bound constrained global optimization. Optim. Methods Softw. 27, 1059–1072 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Price, C.J., Reale, M., Robertson, B.L.: One side cut accelerated random search—A direct search method for bound constrained global optimization. Optim. Lett. 8, 1137–1148 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Radulovia, D.: Pure random search with exponential rate of convergency. Optimization 59(2), 289–303 (2010)

    Article  MathSciNet  Google Scholar 

  20. Rinnooy Kan, A.H.G., Timmer, G.T.: Stochastic global optimization methods part I: clustering methods. Math. Progr. 39, 27–56 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rinnooy Kan, A.H.G., Timmer, G.T.: Stochastic global optimizationmethods part II: multi-level methods. Math. Progr. 39, 57–78 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rubinstein, R.Y.: The cross-entropy method for combinatorial and continuous optimization. Methodol. Comput. Appl. Probab. 2, 127–190 (1999)

    Article  Google Scholar 

  23. Rubinstein, R.Y., Kroese, D.P.: The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning. Springer, New York (2004)

    Book  Google Scholar 

  24. Schoen, F.: Stochastic techniques for global optimization: a survey of recent advances. J. Global Optim. 1(3), 207–228 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tang, Z.B.: Adaptive partitioned random search to global optimization. IEEE Trans. Autom. Control 11, 2235–2244 (1994)

    Article  Google Scholar 

  26. Thomas, W.: Global optimization algorithms – theory and application. http://www.it-weise.de/

  27. Vavasis, S.A.: Complexity issues in global optimization: a survey. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 27–41. Kluwer, The Netherlands (1995)

    Chapter  Google Scholar 

  28. Wu, D.H., Yu, W.Y., Zheng, Q.: A sufficient and necessary condition for global optimization. Appl. Math. Lett. 23(1), 17–21 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yao, Y.R., Chen, L., Zheng, Q.: Optimality condition and algorithm with deviation integral for global optimization. J. Math. Anal. Appl. 357(2), 371–384 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zabinsky, Z.B., Smith, R.L.: Pure adaptive search in global optimization. Math. Progr. 53, 323–338 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zabinsky, Z.B., Wood, G.R., Steel, M.A., Baritompa, W.P.: Pure adaptive search for finite global optimization. Math. Progr. 69, 443–448 (1995)

    MathSciNet  MATH  Google Scholar 

  32. Zheng, Q.: Robust analsys and global optimization. Ann. Oper. Res. 24, 273–286 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zheng, Q., Zhuang, D.M.: Integral global minimization: algorithms, implementations and numerical tests. J. Glob. Optim. 7, 421–454 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Zheng Peng

  2. Department of Mathematics, College of Science, Shanghai University, Shanghai, 200444, China

    Donghua Wu

  3. Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou, 350108, China

    Wenxing Zhu

Authors
  1. Zheng Peng
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  2. Donghua Wu
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  3. Wenxing Zhu
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Corresponding author

Correspondence to Zheng Peng.

Additional information

This work was supported by the Natural Science Foundation of China (61170308), Major Science Foundation of Fujian Provincial Department of Education (JA14037) and talent foundation of Fuzhou University (XRC-1043).

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Peng, Z., Wu, D. & Zhu, W. The robust constant and its applications in random global search for unconstrained global optimization. J Glob Optim 64, 469–482 (2016). https://doi.org/10.1007/s10898-014-0256-1

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  • DOI: https://doi.org/10.1007/s10898-014-0256-1

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