Parallel Dimensionality Reduction for Multiextremal Optimization Problems

  • Victor Gergel
  • Vladimir GrishaginEmail author
  • Ruslan Israfilov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11657)


The paper is devoted to consideration of numerical global optimization methods in the framework of the approach of reducing dimensionality based on nested optimization schemes. For the adaptive nested scheme being more efficient in comparison with its classical prototype a new algorithm of parallel implementation is proposed. General descriptions of the parallel techniques both for synchronous and asynchronous versions are given. Results of numerical experiments on a set of complicated multiextremal test problems of high dimension are presented. These results demonstrate essential acceleration of asynchronous parallel algorithm in comparison with the sequential version.


Multiextremal optimization Global optimum Dimensionality reduction Parallel algorithms 



The research has been supported by the Russian Science Foundation, project No 16-11-10150 “Novel efficient methods and software tools for time-consuming decision make problems using superior-performance supercomputers”.


  1. 1.
    Androulakis, I.P., Floudas, C.A.: Distributed branch and bound algorithms for global optimization. In: Pardalos, P.M. (ed.) Parallel Processing of Discrete Problems. The IMA Volumes in Mathematics and its Applications, vol. 106, pp. 1–35. Springer, New York (1999). Scholar
  2. 2.
    Barkalov, K., Gergel, V.: Parallel global optimization on GPU. J. Glob. Optim. 66(1), 3–20 (2016). Scholar
  3. 3.
    Bartholomew-Biggs, M., Parkhurst, S., Wilson, S.: Using direct to solve anaircraft routing problem. Comput. Optim. Appl. 21(3), 311–323 (2002). Scholar
  4. 4.
    Butz, A.R.: Space-filling curves and mathematical programming. Inform. Control 12, 314–330 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carr, C.R., Howe, C.W.: Quantitative Decision Procedures in Management and Economic: Deterministic Theory and Applications. McGraw-Hill, New York (1964)Google Scholar
  6. 6.
    Dam, E.R., Husslage, B., Hertog, D.: One-dimensional nested maximin designs. J. Glob. Optim. 46, 287–306 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dean, J., Ghemawat, S.: MapReduce: simplified data processing on large clusters. In: Sixth Symposium on Operating System Design and Implementation, OSDI 2004, San Francisco, CA, pp. 137–150 (2004)Google Scholar
  8. 8.
    Evtushenko, Y.G., Malkova, V.U., Stanevichyus, A.A.: Parallel globaloptimization of functions of several variables. Comput. Math. Math. Phys. 49(2), 246–260 (2009). Scholar
  9. 9.
    Famularo, D., Pugliese, P., Sergeyev, Y.: A global optimization technique for checking parametric robustness. Automatica 35, 1605–1611 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gaviano, M., Kvasov, D.E., Lera, D., Sergeyev, Y.D.: Software for generation ofclasses of test functions with known local and global minima for globaloptimization. ACM Trans. Math. Softw. 29(4), 469–480 (2003)CrossRefGoogle Scholar
  11. 11.
    Gergel, V.P., Grishagin, V.A., Gergel, A.V.: Adaptive nested optimization scheme for multidimensional global search. J. Glob. Optim. 66, 35–51 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gergel, V.P., Grishagin, V.A., Israfilov, R.A.: Local tuning in nested scheme of global optimization. Proc. Comput. Sci. 51, 865–874 (2015)CrossRefGoogle Scholar
  13. 13.
    Gergel, V.P., Kuzmin, M.I., Solovyov, N.A., Grishagin, V.A.: Recognition of surface defects of cold-rolling sheets based on method of localities. Int. Rev. Autom. Control 8, 51–55 (2015)Google Scholar
  14. 14.
    Goertzel, B.: Global optimization with space-filling curves. Appl. Math. Lett. 12, 133–135 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grishagin, V.A., Israfilov, R.A.: Multidimensional constrained global optimization in domains with computable boundaries. In: CEUR Workshop Proceedings, vol. 1513, pp. 75–84 (2015)Google Scholar
  16. 16.
    Grishagin, V.A., Israfilov, R.A.: Global search acceleration in the nested optimization scheme. In: AIP Conference Proceedings, vol. 1738, p. 400010 (2016)Google Scholar
  17. 17.
    Grishagin, V.A., Sergeyev, Y.D., Strongin, R.G.: Parallel characteristical algorithms for solving problems of global optimization. J. Glob. Optim. 10, 185–206 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grishagin, V.: On convergence conditions for a class of global search algorithms. In: Proceedings of the 3-rd All-Union Seminar Numerical Methods of Nonlinear Programming, pp. 82–84 (1979, in Russian)Google Scholar
  19. 19.
    Grishagin, V., Israfilov, R., Sergeyev, Y.: Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes. Appl. Math. Comput. 318, 270–280 (2018). Recent Trends in Numerical Computations: Theory and AlgorithmsMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    He, J., Verstak, A., Watson, L.T., Sosonkina, M.: Design and implementation of a massively parallel version of DIRECT. Comput. Optim. Appl. 40(2), 217–245 (2008). Scholar
  21. 21.
    Herrera, J.F.R., Salmerón, J.M.G., Hendrix, E.M.T., Asenjo, R., Casado, L.G.: On parallel branch and bound frameworks for global optimization. J. Glob. Optim. 69(3), 547–560 (2017). Scholar
  22. 22.
    Hime, A., Oliveira Jr., H., Petraglia, A.: Global optimization using space-filling curves and measure-preserving transformations. Soft Comput. Industr. Appl. 96, 121–130 (2011)CrossRefGoogle Scholar
  23. 23.
    Horst, R., Pardalos, P.M.: Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)CrossRefGoogle Scholar
  24. 24.
    Jones, D.R.: The DIRECT global optimization algorithm. In: Floudas, C., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 431–440. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  25. 25.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kvasov, D.E., Menniti, D., Pinnarelli, A., Sergeyev, Y.D., Sorrentino, N.: Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions. Electr. Power Syst. Res. 78, 1217–1229 (2008)CrossRefGoogle Scholar
  27. 27.
    Kvasov, D.E., Pizzuti, C., Sergeyev, Y.D.: Local tuning and partition strategies for diagonal GO methods. Numer. Math. 94, 93–106 (2003)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lera, D., Sergeyev, Y.D.: Lipschitz and Hölder global optimization using space-filling curves. Appl. Numer. Math. 60, 115–129 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lera, D., Sergeyev, Y.D.: Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and holder constants. Commun. Nonlinear Sci. Numer. Simul. 23, 328–342 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Modorskii, V.Y., Gaynutdinova, D.F., Gergel, V.P., Barkalov, K.A.: Optimization in design of scientific products for purposes of cavitation problems. In: AIP Conference Proceedings, vol. 1738, p. 400013 (2016)Google Scholar
  31. 31.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer, NewYork (2014). Scholar
  32. 32.
    Pintér, J.D.: Global Optimization in Action. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  33. 33.
    Sergeyev, Y.D., Grishagin, V.A.: Parallel asynchronous global search and the nested optimization scheme. J. Comput. Anal. Appl. 3, 123–145 (2001)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York (2013). Scholar
  35. 35.
    Sergeyev, Y., Kvasov, D.: Deterministic Global Optimization: An Introduction to the Diagonal Approach. Springer, New York (2017). Scholar
  36. 36.
    Shevtsov, I.Y., Markine, V.L., Esveld, C.: Optimal design of wheel profile for railway vehicles. In: Proceedings of the 6th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Gothenburg, Sweden, pp. 231–236 (2003)Google Scholar
  37. 37.
    Shi, L., Ólafsson, S.: Nested partitions method for global optimization. Oper. Res. 48, 390–407 (2000)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Strongin, R.G.: Numerical Methods in Multiextremal Problems (Information-Statistical Algorithms). Nauka, Moscow (1978, in Russian)Google Scholar
  39. 39.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers/Springer, Dordrecht/Heiselberg (2014)zbMATHGoogle Scholar
  40. 40.
    Sysoyev, A., Barkalov, K., Sovrasov, V., Lebedev, I., Gergel, V.: Globalizer – a parallel software system for solving global optimization problems. In: Malyshkin, V. (ed.) PaCT 2017. LNCS, vol. 10421, pp. 492–499. Springer, Cham (2017). Scholar
  41. 41.
    White, T.: Hadoop: The Definitive Guide. O’Reilly Media, Inc., Newton (2009)Google Scholar
  42. 42.
    Zaharia, M., Chowdhury, M., Franklin, M.J., Shenker, S., Stoica, I.: Spark: cluster computing with working sets. In: Proceedings of the 2Nd USENIX Conference on Hot Topics in Cloud Computing, HotCloud 2010, p. 10. USENIX Association, Berkeley (2010).
  43. 43.
    Zhao, Zh., Meza, J.C., Van Hove, M.: Using pattern search methods for surface structure determination of nanomaterials. J. Phys.: Condens. Matter 18(39), 8693–8706 (2006) Google Scholar
  44. 44.
    Zhigljavsky, A.A., Žilinskas, A.: Stochastic Global Optimization. Springer, NewYork (2008). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Lobachevsky State UniversityNizhni NovgorodRussia

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