Journal of Global Optimization

, Volume 67, Issue 1–2, pp 425–450 | Cite as

Application of Reduced-set Pareto-Lipschitzian Optimization to truss optimization

  • Jonas Mockus
  • Remigijus Paulavičius
  • Dainius Rusakevičius
  • Dmitrij Šešok
  • Julius ŽilinskasEmail author


In this paper, a recently proposed global Lipschitz optimization algorithm Pareto-Lipschitzian Optimization with Reduced-set (PLOR) is further developed, investigated and applied to truss optimization problems. Partition patterns of the PLOR algorithm are similar to those of DIviding RECTangles (DIRECT), which was widely applied to different real-life problems. However here a set of all Lipschitz constants is reduced to just two: the maximal and the minimal ones. In such a way the PLOR approach is independent of any user-defined parameters and balances equally local and global search during the optimization process. An expanded list of other well-known DIRECT-type algorithms is used in investigation and experimental comparison using the standard test problems and truss optimization problems. The experimental investigation shows that the PLOR algorithm gives very competitive results to other DIRECT-type algorithms using standard test problems and performs pretty well on real truss optimization problems.


Truss optimization Lipschitz optimization PLOR algorithm DIRECT algorithm 



This research was funded by a grant (No. MIP-051/2014) from the Research Council of Lithuania.


  1. 1.
    Bartholomew-Biggs, M.C., Parkhurst, S.C., Wilson, S.P.: Using DIRECT to solve an aircraft routing problem. Comput. Optim. Appl. 21(3), 311–323 (2002). doi: 10.1023/A:1013729320435 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carter, R.G., Gablonsky, J.M., Patrick, A., Kelley, C.T., Eslinger, O.J.: Algorithms for noisy problems in gas transmission pipeline optimization. Optim. Eng. 2(2), 139–157 (2001). doi: 10.1023/A:1013123110266 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Choi, T.D., Eslinger, O.J., Gilmore, P., Patrick, A., Kelley, C.T., Gablonsky, J.M.: Iffco: implicit filtering for constrained optimization, version 2. Rep. CRSC-TR99, 23 (1999)Google Scholar
  4. 4.
    Cox, S.E., Haftka, R.T., Baker, C.A., Grossman, B., Mason, W.H., Watson, L.T.: A comparison of global optimization methods for the design of a high-speed civil transport. J. Glob. Optim. 21(4), 415–432 (2001). doi: 10.1023/A:1012782825166 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Deb, K., Gulati, S.: Design of truss-structures for minimum weight using genetic algorithms. Finite Elem. Anal. Des. 37(5), 447–465 (2001). doi: 10.1016/S0168-874X(00)00057-3 CrossRefzbMATHGoogle Scholar
  6. 6.
    Figueira, J., Greco, S., Ehrgott, M.: Multiple Criteria Decision Analysis: State of the Art Surveys. Springer, Berlin (2004)zbMATHGoogle Scholar
  7. 7.
    Finkel, D.E.: DIRECT optimization algorithm user guide. Technical report, Center for Research in Scientific Computation. North Carolina State University, Raleigh, NC (2003)Google Scholar
  8. 8.
    Finkel, D.E.: Global optimization with the DIRECT algorithm. Ph.D. thesis, North Carolina State University (2005)Google Scholar
  9. 9.
    Finkel, D.E., Kelley, C.T.: Additive scaling and the DIRECT algorithm. J. Glob. Optim. 36, 597–608 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gablonsky, J.M.: Modifications of the DIRECT algorithm. Ph.D. thesis, North Carolina State University, Raleigh, NC (2001)Google Scholar
  11. 11.
    Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Glob. Optim. 21, 27–37 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grbić, R., Nyarko, E.K., Scitovski, R.: A modification of the DIRECT method for Lipschitz global optimization for a symmetric function. J. Glob. Optim. 57(4), 1193–1212 (2013). doi: 10.1007/s10898-012-0020-3 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    He, J., Verstak, A.A., Watson, L.T., Stinson, C.A., Ramakrishnan, N., Shaffer, C.A., Rappaport, T.S., Anderson, C.R., Bae, K.K., Jiang, J., et al.: Globally optimal transmitter placement for indoor wireless communication systems. IEEE Trans. Wirel. Commun. 3(6), 1906–1911 (2004)CrossRefGoogle Scholar
  14. 14.
    Jones, D.R.: The DIRECT global optimization algorithm. In: Floudas, C.A., Pardalos, P.M. (eds.) The Encyclopedia of Optimization, pp. 431–440. Kluwer, Dordrect (2001)CrossRefGoogle Scholar
  15. 15.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kvasov, D.E., Sergeyev, Y.D.: A univariate global search working with a set of Lipschitz constants for the first derivative. Optim. Lett. 3(2), 303–318 (2009). doi: 10.1007/s11590-008-0110-9 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kvasov, D.E., Sergeyev, Y.D.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236(16), 4042–4054 (2012). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lera, D., Sergeyev, Y.D.: Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Hölder constants. Commun. Nonlinear Sci. Numer. Simul. 23(1–3), 328–342 (2015). doi: 10.1016/j.cnsns.2014.11.015 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, J.P.: Truss topology optimization using an improved species-conserving genetic algorithm. Eng. Optim. 47(1), 107–128 (2015). doi: 10.1080/0305215X.2013.875165 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, L.J., Huang, Z.B., Liu, F., Wu, Q.H.: A heuristic particle swarm optimizer for optimization of pin connected structures. Comput. Struct. 85(7), 340–349 (2007)CrossRefGoogle Scholar
  21. 21.
    Li, Y., Peng, Y., Zhou, S.: Improved pso algorithm for shape and sizing optimization of truss structure. J. Civ. Eng. Manag. 19(4), 542–549 (2013)CrossRefGoogle Scholar
  22. 22.
    Liu, Q.: Linear scaling and the DIRECT algorithm. J. Glob. Optim. 56(3), 1233–1245 (2013). doi: 10.1007/s10898-012-9952-x MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, Q., Cheng, W.: A modified DIRECT algorithm with bilevel partition. J. Glob. Optim. 60(3), 483–499 (2014). doi: 10.1007/s10898-013-0119-1 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems. Comput. Optim. Appl. 45, 353–375 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Glob. Optim. 48, 113–128 (2010). doi: 10.1007/s10898-009-9515-y MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lu, Y.C., Jan, J.C., Hung, S.L., Hung, G.H.: Enhancing particle swarm optimization algorithm using two new strategies for optimizing design of truss structures. Eng. Optim. 45(10), 1251–1271 (2013). doi: 10.1080/0305215X.2012.729054 CrossRefGoogle Scholar
  27. 27.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  28. 28.
    Mockus, J.: On the Pareto pptimality in the context of Lipschitzian optimization. Informatica 22(4), 521–536 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mockus, J., Paulavičius, R.: On the reduced-set Pareto–Lipschitzian optimization. Comput. Sci. Tech. 1(2), 184–192 (2013)CrossRefGoogle Scholar
  30. 30.
    Pardalos, P.M., Siskos, Y. (eds.): Advances in Multi-criteria Analysis. Kluwer, Dordrecht (1995)Google Scholar
  31. 31.
    Paulavičius, R., Sergeyev, Y.D., Kvasov, D.E., Žilinskas, J.: Globally-biased Disimpl algorithm for expensive global optimization. J. Glob. Optim. 59(2–3), 545–567 (2014). doi: 10.1007/s10898-014-0180-4 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Paulavičius, R., Žilinskas, J.: Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints. Optim. Lett. (2014). doi: 10.1007/s11590-014-0772-4 zbMATHGoogle Scholar
  33. 33.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization Springer Briefs in Optimization. Springer, New York (2014). doi: 10.1007/978-1-4614-9093-7 CrossRefzbMATHGoogle Scholar
  34. 34.
    Paulavičius, R., Žilinskas, J.: Simplicial Lipschitz optimization without the Lipschitz constant. J. Glob. Optim. 59(1), 23–40 (2014). doi: 10.1007/s10898-013-0089-3 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Perez, R., Behdinan, K.: Particle swarm approach for structural design optimization. Comput. Struct. 85(19), 1579–1588 (2007)CrossRefGoogle Scholar
  36. 36.
    Schmit, L.A., Farshi, B.: Some approximation concepts for structural synthesis. AIAA J. 12(5), 692–699 (1974)CrossRefGoogle Scholar
  37. 37.
    Sergeyev, Y.D., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16(3), 910–937 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tang, H., Li, F., Wang, Y., Xue, S., Cheng, R.: Particle swarm optimization algorithm for shape optimization of truss structures. J. Harbin Inst. Technol. 41(12), 94–99 (2009)Google Scholar
  39. 39.
    Zhu, H., Bogy, D.B.: DIRECT algorithm and its application to slider air-bearing surface optimization. IEEE Trans. Magn. 38(5), 2168–2170 (2002)CrossRefGoogle Scholar
  40. 40.
    Zhu, H., Bogy, D.B.: Hard disc drive air bearing design: modified DIRECT algorithm and its application to slider air bearing surface optimization. Tribol. Int. 37(2), 193–201 (2004)CrossRefGoogle Scholar
  41. 41.
    Žilinskas, A.: On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl. Math. Comput. 218(16), 8131–8136 (2012). doi: 10.1016/j.amc.2011.07.051 MathSciNetzbMATHGoogle Scholar
  42. 42.
    Žilinskas, J., Kvasov, D.E., Paulavičius, R., Sergeyev, Y.D.: Acceleration of simplicial-partition-based methods in Lipschitz global optimization. In: Gergel, V.P. (ed.) High-Performance Computing on Clusters, pp. 128–133. Nizhny Novgorod State University, Nizhny Novgorod (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jonas Mockus
    • 1
  • Remigijus Paulavičius
    • 1
  • Dainius Rusakevičius
    • 2
  • Dmitrij Šešok
    • 2
  • Julius Žilinskas
    • 1
    Email author
  1. 1.Vilnius University Institute of Mathematics and InformaticsVilniusLithuania
  2. 2.Vilnius Gediminas Technical UniversityVilniusLithuania

Personalised recommendations