Journal of Global Optimization

, Volume 59, Issue 1, pp 23–40 | Cite as

Simplicial Lipschitz optimization without the Lipschitz constant

  • Remigijus Paulavičius
  • Julius ŽilinskasEmail author


In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant. Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Almost all previous modifications of Direct algorithm use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. Simplicial partitions may be preferable when the initial feasible region is either already a simplex or may be covered by one or a manageable number of simplices. Therefore in this paper we propose and investigate simplicial versions of the partition-based algorithm. In the case of simplicial partitions, definition of potentially optimal subregion cannot be the same as in the rectangular version. In this paper we propose and investigate two definitions of potentially optimal simplices: one involves function values at the vertices of the simplex and another uses function value at the centroid of the simplex. We use experimental investigation to compare performance of the algorithms with different definitions of potentially optimal partitions. The experimental investigation shows, that proposed simplicial algorithm gives very competitive results to Direct algorithm using standard test problems and performs particularly well when the search space and the numbers of local and global optimizers may be reduced by taking into account symmetries of the objective function.


Global optimization Lipschitz optimization Simplicial Direct-type algorithm Potentially optimal simplex 



The authors would like to thank anonymous referees for their careful reading of the paper and insightful comments that helped us improve the paper. Postdoctoral fellowship of R. Paulavičius is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania” within the framework of the Measure for Enhancing Mobility of Scholars and Other Researchers and the Promotion of Student Research (VP1-3.1-ŠMM-01) of the Program of Human Resources Development Action Plan.


  1. 1.
    Baker, C.A., Watson, L.T., Grossman, B., Mason, W.H., Haftka, R.T.: Parallel global aircraft configuration design space exploration. In: Tentner, A. (ed.) High Performance Computing Symposium 2000, pp. 54–66. Soc. for Computer Simulation Internat (2000)Google Scholar
  2. 2.
    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 CrossRefGoogle Scholar
  3. 3.
    Björkman, M., Holmström, K.: Global optimization using the direct algorithm in Matlab. Adv. Model. Optim. 1(2), 17–37 (1999)Google Scholar
  4. 4.
    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 CrossRefGoogle Scholar
  5. 5.
    Casado, L., Hendrix, E., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. J. Glob. Optim. 39(4), 577–593 (2007). doi: 10.1007/s10898-007-9157-x CrossRefGoogle Scholar
  6. 6.
    Chiter, L.: DIRECT algorithm: a new definition of potentially optimal hyperrectangles. Appl. Math. Comput. 179(2), 742–749 (2006). doi: 10.1016/j.amc.2005.11.127 CrossRefGoogle Scholar
  7. 7.
    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 CrossRefGoogle Scholar
  8. 8.
    Dixon, L., Szegö, C.: The global optimisation problem: an introduction. In: Dixon, L., Szegö, G. (eds.) Towards Global Optimization, vol. 2, pp. 1–15. North-Holland Publishing Company, Amsterdam (1978)Google Scholar
  9. 9.
    Finkel, D.E., Kelley, C.T.: Additive scaling and the DIRECT algorithm. J. Glob. Optim. 36(4), 597–608 (2006). doi: 10.1007/s10898-006-9029-9 CrossRefGoogle Scholar
  10. 10.
    Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Glob. Optim. 21(1), 27–37 (2001). doi: 10.1023/A:1017930332101 CrossRefGoogle Scholar
  11. 11.
    Gorodetsky, S.: Paraboloid triangulation methods in solving multiextremal optimization problems with constraints for a class of functions with Lipschitz directional derivatives. Vestnik of Lobachevsky State University of Nizhni Novgorod 1(1), 144–155 (2012)Google 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. 1–20 (2012). doi: 10.1007/s10898-012-0020-3
  13. 13.
    He, J., Watson, L.T., Ramakrishnan, N., Shaffer, C.A., Verstak, A., Jiang, J., Bae, K., Tranter, W.H.: Dynamic data structures for a DIRECT search algorithm. Comput. Optim. Appl. 23(1), 5–25 (2002). doi: 10.1023/A:1019992822938 CrossRefGoogle Scholar
  14. 14.
    Horst, R.: On generalized bisection of n-simplices. Math. Comput. 66(218), 691–698 (1997)CrossRefGoogle Scholar
  15. 15.
    Horst, R.: Bisecton by global optimization revisited. J. Optim. Theory Appl. 144(3), 501–510 (2010)CrossRefGoogle Scholar
  16. 16.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and Its Application. Kluwer Academic Publishers, Boston (1995)Google Scholar
  17. 17.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)CrossRefGoogle Scholar
  18. 18.
    Jennrich, R.I., Sampson, P.F.: Application of stepwise regression to non-linear estimation. Technometrics 10(1), 63–72 (1968). doi: 10.1080/00401706.1968.10490535 CrossRefGoogle Scholar
  19. 19.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993). doi: 10.1007/BF00941892 CrossRefGoogle Scholar
  20. 20.
    Křivý, I., Tvrdík, J., Krpec, R.: Stochastic algorithms in nonlinear regression. Comput. Stat. Data Anal. 33(3), 277–290 (2000). doi: 10.1016/S0167-9473(99)00059-6 CrossRefGoogle Scholar
  21. 21.
    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 CrossRefGoogle Scholar
  22. 22.
    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/ CrossRefGoogle Scholar
  23. 23.
    Kvasov, D.E., Sergeyev, Y.D.: Univariate geometric Lipschitz global optimization algorithms. Numer. Algebra Control Optim. 2(1), 69–90 (2012). doi: 10.3934/naco.2012.2.69 CrossRefGoogle Scholar
  24. 24.
    Lanczos, C.: Applied Analysis. Prentice Hall, Englewood Cliffs (1956)Google Scholar
  25. 25.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A direct-based approach for large-scale global optimization problems. Comput. Optim. Appl. 45(2), 353–375 (2010)CrossRefGoogle Scholar
  26. 26.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Global Optim. 48(1), 113–128 (2010). doi: 10.1007/s10898-009-9515-y CrossRefGoogle Scholar
  27. 27.
    Mockus, J.: On the Pareto optimality in the context of Lipschitzian optimization. Informatica 22(4), 521–536 (2011)Google Scholar
  28. 28.
    Nast, M.: Subdivision of simplices relative to a cutting plane and finite concave minimization. J. Global Optim. 9(1), 65–93 (1996). doi: 10.1007/BF00121751 CrossRefGoogle Scholar
  29. 29.
    Osborne, M.R.: Some aspects of nonlinear least squares calculations. In: Lootsma, F.A. (ed.) Numerical Methods for Nonlinear Optimization, pp. 171–189. Academic Press, New York (1972)Google Scholar
  30. 30.
    Paulavičius, R., Žilinskas, J.: Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Inf. Technol. Control 36(4), 383–387 (2007)Google Scholar
  31. 31.
    Paulavičius, R., Žilinskas, J.: Influence of Lipschitz bounds on the speed of global optimization. Technol. Econ. Dev. Econ. 18(1), 54–66 (2012). doi: 10.3846/20294913.2012.661170 CrossRefGoogle Scholar
  32. 32.
    Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim. Lett. 4(2), 173–183 (2010). doi: 10.1007/s11590-009-0156-3 CrossRefGoogle Scholar
  33. 33.
    Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Springer, New York (1996)Google Scholar
  34. 34.
    di Serafino, D., Liuzzi, G., Piccialli, V., Riccio, F., Toraldo, G.: A modified DIviding RECTangles algorithm for a problem in astrophysics. J. Optim. Theory Appl. 151(1), 175–190 (2011). doi: 10.1007/s10957-011-9856-9 CrossRefGoogle Scholar
  35. 35.
    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). doi: 10.1137/040621132 CrossRefGoogle Scholar
  36. 36.
    Sergeyev, Y.D., Kvasov, D.E.: Lipschitz global optimization. In: Cochran, J. (ed.) Wiley Encyclopedia of Operations Research and Management Science, vol. 4, pp. 2812–2828. Wiley, New York (2011)Google Scholar
  37. 37.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  38. 38.
    Todt, M.J.: The Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems, vol. 24 (1976)Google Scholar
  39. 39.
    Yao, Y.: Dynamic tunneling algorithm for global optimization. IEEE Trans. Syst. Man Cybern. 19(5), 1222–1230 (1989)CrossRefGoogle Scholar
  40. 40.
    Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)Google 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 CrossRefGoogle Scholar
  42. 42.
    Žilinskas, A., Žilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. Comput. Math. Appl. 44(7), 957–967 (2002). doi: 10.1016/S0898-1221(02)00206-7 CrossRefGoogle Scholar
  43. 43.
    Žilinskas, A., Žilinskas, J.: A hybrid global optimization algorithm for non-linear least squares regression. J. Global Optim. 56(2), 265–277 (2013). doi: 10.1007/s10898-011-9840-9
  44. 44.
    Žilinskas, J.: Reducing of search space of multidimensional scaling problems with data exposing symmetries. Inf. Technol. Control 36(4), 377–382 (2007)Google Scholar
  45. 45.
    Žilinskas, J.: Branch and bound with simplicial partitions for global optimization. Math. Modell. Anal. 13(1), 145–159 (2008). doi: 10.3846/1392-6292.2008.13.145-159 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Vilnius University Institute of Mathematics and InformaticsVilniusLithuania

Personalised recommendations