Skip to main content

Lipschitz Optimization with Different Bounds over Simplices

  • Chapter
  • First Online:
Simplicial Global Optimization

Part of the book series: SpringerBriefs in Optimization ((BRIEFSOPTI))

  • 1051 Accesses

Abstract

Many problems in engineering, physics, economics, and other fields may be formulated as optimization problems, where the optimal value of an objective function must be found [23, 55, 59, 110, 114, 134, 136].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baravykaitė, M., Čiegis, R., Žilinskas, J.: Template realization of generalized branch and bound algorithm. Math. Model. Anal. 10(3), 217–236 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Baritompa, W.: Customizing methods for global optimization — a geometric viewpoint. J. Global Optim. 3(2), 193–212 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Breiman, L., Cutler, A.: A deterministic algorithm for global optimization. Math. Program. 58(1–3), 179–199 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Butz, A.R.: Space filling curves and mathematical programming. Inform. Contr. 12, 319–330 (1968)

    Article  MathSciNet  Google Scholar 

  5. Chandra, R., Menon, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J.: Parallel Programming in OpenMP. Morgan Kaufmann, Los Altos (2000)

    Google Scholar 

  6. Chapman, B., Jost, G., Van Der Pas, R.: Using OpenMP: Portable Shared Memory Parallel Programming, vol. 10. MIT, Cambridge (2008)

    Google Scholar 

  7. Čiegis, R., Henty, D., Kågström, B., Žilinskas, J. (eds.): Parallel Scientific Computing and Optimization. Springer Optimization and Its Applications, vol. 27. Springer, New York (2009)

    Google Scholar 

  8. D’Apuzzo, M., Marino, M., Migdalas, A., Pardalos, P.M., Toraldo, G.: Parallel computing in global optimization. In: Kontoghiorghes, E.J. (ed.) Handbook of Parallel Computing and Statistics, pp. 225–258. Chapman & Hall, London (2006)

    Google Scholar 

  9. Evtushenko, Y., Posypkin, M.: A deterministic approach to global box-constrained optimization. Optim. Lett. 7(4), 819–829 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferreira, A., Pardalos, P.M. (eds.): Solving Combinatorial Optimization Problems in Parallel: Methods and Techniques. Lecture Notes in Computer Science, vol. 1054. Springer, New York (1996)

    Google Scholar 

  11. Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, vol. 1–6. Kluwer, Dordrecht (2001)

    Book  MATH  Google Scholar 

  12. Galperin, E.A.: The cubic algorithm. J. Math. Anal. Appl. 112(2), 635–640 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Galperin, E.A.: Precision, complexity, and computational schemes of the cubic algorithm. J. Optim. Theor. Appl. 57, 223–238 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gendron, B., Crainic, T.G.: Parallel branch-and-bound algorithms: survey and synthesis. Oper. Res. 42(6), 1042–1066 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gergel, V.P.: A global optimization algorithm for multivariate function with Lipschitzian first derivatives. J. Global Optim. 10(3), 257–281 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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, 144–155 (2012)

    Google Scholar 

  17. Gourdin, E., Hansen, P., Jaumard, B.: Global optimization of multivariate Lipschitz functions: Survey and computational comparison. Les Cahiers du GERAD (1994)

    Google Scholar 

  18. Gropp, W., Lusk, E.L., Skjellum, A.: Using MPI-: Portable Parallel Programming with the Message Passing Interface, vol. 1. MIT, Cambridge (1999)

    Google Scholar 

  19. Hansen, P., Jaumard, B.: Lipshitz optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, vol. 1, pp. 407–493. Kluwer, Dordrecht (1995)

    Chapter  Google Scholar 

  20. Horst, R.: A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization. J. Optim. Theor. Appl. 51, 271–291 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and Its Application. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  22. Horst, R., Tuy, H.: On the convergence of global methods in multiextremal optimization. J. Optim. Theor. Appl. 54, 253–271 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  24. Jansson, C., Knuppel, O.: A global minimization method: The multi-dimensional case. Tech. rep., TU Hamburg-Harburg (1992)

    Google Scholar 

  25. Jaumard, B., Ribault, H., Herrmann, T.: An on-line cone intersection algorithm for global optimization of multivariate Lipschitz functions. Cahiers du GERAD 95(7) (1995)

    Google Scholar 

  26. Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theor. Appl. 79(1), 157–181 (1993). doi:10.1007/BF00941892

    Article  MathSciNet  MATH  Google Scholar 

  27. Kolmogorov, A., Fomin, S.: Elements of Function Theory and Functional Analysis. Nauka, Moscow (1968)

    Google Scholar 

  28. Kvasov, D.E., Pizzuti, C., Sergeyev, Y.D.: Local tuning and partition strategies for diagonal go methods. Numerische Mathematik 94(1), 93–106 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kvasov, D.E., Sergeyev, Y.D.: Multidimensional global optimization algorithm based on adaptive diagonal curves. Comput. Math. Math. Phys. 43(1), 40–56 (2003)

    MathSciNet  Google Scholar 

  30. Kvasov, D.E., Sergeyev, Y.D.: A univariate global search working with a set of Lipschitz constants for the first derivative. Optim. Lett. 3, 303–318 (2009). doi:10.1007/s11590-008-0110-9

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lera, D., Sergeyev, Y.D.: Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives. SIAM J. Optim. 23(1), 508–529 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Madsen, K., Žilinskas, J.: Testing branch-and-bound methods for global optimization. Tech. Rep. IMM-REP-2000-05, Technical University of Denmark (2000)

    Google Scholar 

  34. Mayne, D.Q., Polak, E.: Outer approximation algorithm for nondifferentiable optimization problems. J. Optim. Theor. Appl. 42(1), 19–30 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Meewella, C.C., Mayne, D.Q.: An algorithm for global optimization of Lipschitz continuous functions. J. Optim. Theor. Appl. 57(2), 307–323 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Meewella, C.C., Mayne, D.Q.: An efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions. J. Optim. Theor. Appl. 61(2), 247–270 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  37. Migdalas, A., Pardalos, P.M., Storøy, S.: Parallel Computing in Optimization. Applied Optimization, vol. 7. Kluwer, Dordrecht (1997)

    Google Scholar 

  38. Mladineo, R.H.: An algorithm for finding the global maximum of a multimodal, multivariate function. Math. Program. 34(2), 188–200 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mladineo, R.H.: Convergence rates of a global optimization algorithm. Math. Program. 54(1–3), 223–232 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pardalos, P.M. (ed.): Parallel Processing of Discrete Problems. IMA Volumes in Mathematics and its Applications, vol. 106. Springer, New York (1999)

    Google Scholar 

  41. Paulavičius, R., Žilinskas, J.: Analysis of different norms and corresponding Lipschitz constants for global optimization. Tech. Econ. Dev. Econ. 12(4), 301–306 (2006)

    Google Scholar 

  42. Paulavičius, R., Žilinskas, J.: Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Inform. Tech. Contr. 36(4), 383–387 (2007)

    Google Scholar 

  43. Paulavičius, R., Žilinskas, J.: Improved Lipschitz bounds with the first norm for function values over multidimensional simplex. Math. Model. Anal. 13(4), 553–563 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Paulavičius, R., Žilinskas, J.: Global optimization using the branch-and-bound algorithm with a combination of Lipschitz bounds over simplices. Tech. Econ. Dev. Econ. 15(2), 310–325 (2009)

    Article  Google Scholar 

  45. Paulavičius, R., Žilinskas, J.: Influence of Lipschitz bounds on the speed of global optimization. Tech. Econ. Dev. Econ. 18(1), 54–66 (2012). doi:10.3846/20294913.2012.661170

    Article  Google Scholar 

  46. 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

    Article  MathSciNet  MATH  Google Scholar 

  47. Paulavičius, R., Žilinskas, J., Grothey, A.: Parallel branch and bound for global optimization with combination of Lipschitz bounds. Optim. Meth. Software 26(3), 487–498 (2011)

    Article  Google Scholar 

  48. Pedoe, D.: Circles: A Mathematical View. Math. Assoc. Amer., Washington, DC (1995)

    MATH  Google Scholar 

  49. Pinter, J.: Extended univariate algorithms for n-dimensional global optimization. Computing 36(1), 91–103 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  50. Pinter, J.: Globally convergent methods for n-dimensional multiextremal optimization. Optimization 17, 187–202 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pinter, J.: Branch-and-bound algorithms for solving global optimization problems with Lipschitzian structure. Optimization 19(1), 101–110 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  52. Pinter, J.: Continuous global optimization software: A brief review. Optika 52, 1–8 (1996)

    Google Scholar 

  53. Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Nonconvex Optimization and Its Application. Springer, New York (1996)

    Book  MATH  Google Scholar 

  54. Piyavskii, S.A.: An algorithm for finding the absolute minimum of a function. Theor. Optim. Solut. 2, 13–24 (1967). In Russian

    Google Scholar 

  55. Piyavskii, S.A.: An algorithm for finding the absolute extremum of a function. Zh. Vychisl. Mat. mat. Fiz 12(4), 888–896 (1972)

    Google Scholar 

  56. Sergeyev, Y.D.: An information global optimization algorithm with local tunning. SIAM J. Optim. 5(4), 858–870 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sergeyev, Y.D.: A one-dimensional deterministic global minimization algorithm. Comput. Math. Math. Phys. 35(5), 553–562 (1995)

    MathSciNet  Google Scholar 

  58. Sergeyev, Y.D.: A method using local tuning for minimizing functions with Lipschitz derivatives. In: Developments in Global Optimization, pp. 199–216. Springer, New York (1997)

    Google Scholar 

  59. Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81(1), 127–146 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  60. Sergeyev, Y.D.: Multidimensional global optimization using the first derivatives. Comput. Math. Math. Phys. 39(5), 711–720 (1999)

    MathSciNet  MATH  Google Scholar 

  61. Sergeyev, Y.D.: Univariate global optimization with multiextremal non-differentiable constraints without penalty functions. Comput. Optim. Appl. 34(2), 229–248 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  62. Sergeyev, Y.D., Grishagin, V.: A parallel method for finding the global minimum of univariate functions. J. Optim. Theor. Appl. 80(3), 513–536 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  63. Sergeyev, Y.D., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16, 910–937 (2006). doi:10.1137/040621132

    Article  MathSciNet  MATH  Google Scholar 

  64. Sergeyev, Y.D., Kvasov, D.E.: Diagonal Global Optimization Methods. FizMatLit, Moscow (2008). In Russian

    Google Scholar 

  65. Sergeyev, Y.D., Kvasov, D.E.: Lipschitz global optimization and estimates of the Lipschitz constant. In: Chaoqun, M., Lean, Y., Dabin, Z., Zhongbao, Z. (eds.) Global Optimization: Theory, Methods and Applications, I, pp. 518–521. Global Link, Hong Kong (2009)

    Google Scholar 

  66. Sergeyev, Y.D., Kvasov, D.E., Khalaf, F.M.: A one-dimensional local tuning algorithm for solving go problems with partially defined constraints. Optim. Lett. 1(1), 85–99 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  67. Sergeyev, Y.D., Pugliese, P., Famularo, D.: Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints. Math. Program. 96(3), 489–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  68. Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to global optimization exploiting space-filling curves. Springer, New York (2013)

    Book  MATH  Google Scholar 

  69. Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9, 379–388 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  70. Snir, M.: MPI the Complete Reference: The MPI Core, vol. 1. MIT, Cambridge (1998)

    Google Scholar 

  71. Strongin, R.G.: Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves. J. Global Optim. 2, 357–378 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  72. Strongin, R.G., Sergeyev, Y.D.: Global multidimensional optimization on parallel computer. Parallel Comput. 18(11), 1259–1273 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  73. Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. KAP, Dordrecht (2000)

    Book  Google Scholar 

  74. Wood, G.R.: Multidimensional bisection applied to global optimisation. Comp. Math. Appl. 21(6–7), 161–172 (1991)

    Article  MATH  Google Scholar 

  75. Wood, G.R.: The bisection method in higher dimensions. Math. Program. 55, 319–337 (1992)

    Article  MATH  Google Scholar 

  76. Zhang, B.P., Wood, G., Baritompa, W.: Multidimensional bisection: The performance and the context. J. Global Optim. 3(3), 337–358 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Remigijus Paulavičius, Julius Žilinskas

About this chapter

Cite this chapter

Paulavičius, R., Žilinskas, J. (2014). Lipschitz Optimization with Different Bounds over Simplices. In: Simplicial Global Optimization. SpringerBriefs in Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9093-7_2

Download citation

Publish with us

Policies and ethics