[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Study of multiscale global optimization based on parameter space partition

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Inverse problems in geophysics are usually described as data misfit minimization problems, which are difficult to solve because of various mathematical features, such as multi-parameters, nonlinearity and ill-posedness. Local optimization based on function gradient can not guarantee to find out globally optimal solutions, unless a starting point is sufficiently close to the solution. Some global optimization methods based on stochastic searching mechanisms converge in the limit to a globally optimal solution with probability 1. However, finding the global optimum of a complex function is still a great challenge and practically impossible for some problems so far. This work develops a multiscale deterministic global optimization method which divides definition space into sub-domains. Each of these sub-domains contains the same local optimal solution. Local optimization methods and attraction field searching algorithms are combined to determine the attraction basin near the local solution at different function smoothness scales. With Multiscale Parameter Space Partition method, all attraction fields are to be determined after finite steps of parameter space partition, which can prevent redundant searching near the known local solutions. Numerical examples demonstrate the efficiency, global searching ability and stability of this method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bomze I.M., Csendes T., Horst R., Pardalos P.M.: Developments in Global Optimization: Nonconvex Optimization and Its Applications. Kluwer, Dordrecht (1997)

    Google Scholar 

  2. Gray, P., Hart, W., Painton, L., et al.: A Survey of Global Optimization Methods. Technical report, Sandia National Laboratories (1997)

  3. Scales J.A., Smith M.L., Fischer T.L: Global optimization methods for multimodal inverse problems. J. Comput. Phys. 103, 258–268 (1992)

    Article  Google Scholar 

  4. Deng, L.H., Scales, J.A.: Estimating the Topography of Multi-dimensional Fitness Functions. Colorado School of Mines (1999)

  5. Rothman D.H.: Nonlinear inversion, statistical-mechanics, and residual statics Estimation. Geophysics 50, 2784–2796 (1985)

    Article  Google Scholar 

  6. Rothman D.H.: Automatic estimation of large residual statics corrections. Geophysics 51, 332–346 (1986)

    Article  Google Scholar 

  7. Kirkpatrick S., Gelatt C.D., Vecchi M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  Google Scholar 

  8. Holland J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  9. Goldberg D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989)

    Google Scholar 

  10. Stoffa P.L., Sen M.K.: Nonlinear multiparameter optimization using genetic algorithms—inversion of plane-wave seismograms. Geophysics 56, 1794–1810 (1991)

    Article  Google Scholar 

  11. Sambridge M., Drijkoningen G.: Genetic algorithms in seismic wave-form inversion. Geophys. J. Int. 109, 323–342 (1992)

    Article  Google Scholar 

  12. Gallagher K., Sambridge M., Drijkoningen G.: Genetic algorithms—an evolution from Monte-Carlo methods for strongly nonlinear geophysical optimization problems. Geophys. Res. Lett. 18, 2177–2180 (1991)

    Article  Google Scholar 

  13. Gallagher K., Sambridge M.: Genetic algorithms—a powerful tool for large-scale nonlinear optimization problems. Comput. Geosci. 20, 1229–1236 (1994)

    Article  Google Scholar 

  14. Sen M., Stoffa P.L.: Global Optimization Methods in Geophysical Inversion. Elsevier, Amsterdam (1995)

    Google Scholar 

  15. Gill P.E., Murray W., Wright M.H.: Practical Optimization. Academic Press, New York (1981)

    Google Scholar 

  16. Granville V., Krivanek M., Rasson J.P.: Simulated annealing—a proof of convergence. IEEE Trans. Pattern Anal. Mach. Intell. 16, 652–656 (1994)

    Article  Google Scholar 

  17. Greenhalgh D., Marshall S.: Convergence criteria for genetic algorithms. SIAM J. Comput. 30, 269–282 (2000)

    Article  Google Scholar 

  18. Locatelli M.: Convergence and first hitting time of simulated annealing algorithms for continuous global optimization. Math. Methods Oper. Res. 54, 171–199 (2001)

    Article  Google Scholar 

  19. Locatelli M.: Convergence of a simulated annealing algorithm for continuous global optimization. J. Glob. Optim. 18, 219–234 (2000)

    Article  Google Scholar 

  20. Locatelli M.: Simulated annealing algorithms for continuous global optimization: convergence conditions. J. Optim. Theory Appl. 104, 121–133 (2000)

    Article  Google Scholar 

  21. Locatelli M.: Convergence properties of simulated annealing for continuous global optimization. J. Appl. Probab. 33, 1127–1140 (1996)

    Article  Google Scholar 

  22. Belisle C.J.P.: Convergence theorems for a class of simulated annealing algorithms on R(D). J. Appl. Probab. 29, 885–895 (1992)

    Article  Google Scholar 

  23. Fallat M.R., Dosso S.E.: Geoacoustic inversion via local, global, and hybrid algorithms. J. Acoust. Soc. Am. 105, 3219–3230 (1999)

    Article  Google Scholar 

  24. Liu P.C., Hartzell S., Stephenson W.: Nonlinear multiparameter inversion using a hybrid global search algorithm—applications in reflection seismology. Geophys. J. Int. 122, 991–1000 (1995)

    Article  Google Scholar 

  25. Cary P.W., Chapman C.H.: Automatic 1-D waveform inversion of marine seismic refraction data. Geophys. J. Int. 93, 527–546 (1988)

    Article  Google Scholar 

  26. Gerstoft P.: Inversion of acoustic data using a combination of genetic algorithms and the Gauss–Newton approach. J. Acoust. Soc. Am. 97, 2181–2190 (1995)

    Article  Google Scholar 

  27. Hibbert D.B.: A hybrid genetic algorithm for the estimation of kinetic-parameters. Chemometr. Intell. Lab. 19, 319–329 (1993)

    Article  Google Scholar 

  28. Chunduru R.K., Sen M.K., Stoffa P.L.: Hybrid optimization methods for geophysical inversion. Geophysics 62, 1196–1207 (1997)

    Article  Google Scholar 

  29. Calderon-Macias C., Sen M.K., Stoffa P.L.: Artificial neural networks for parameter estimation in geophysics. Geophys. Prospect. 48, 21–47 (2000)

    Article  Google Scholar 

  30. Chelouah R., Siarry P.: A hybrid method combining continuous tabu search and Nelder–Mead simplex algorithms for the global optimization of multiminima functions. Eur. J. Oper. Res. 161, 636–654 (2005)

    Article  Google Scholar 

  31. Gil C., Marquez A., Banos R. et al.: A hybrid method for solving multi-objective global optimization problems. J. Glob. Optim. 38, 265–281 (2007)

    Article  Google Scholar 

  32. Olensek J., Burmen A., Puhan J., Tuma T.: DESA: a new hybrid global optimization method and its application to analog integrated circuit sizing. J. Glob. Optim. 44, 53–77 (2009)

    Article  Google Scholar 

  33. Yiu K.F.C., Liu Y., Teo K.L.: A hybrid descent method for global optimization. J. Glob. Optim. 28, 229–238 (2004)

    Article  Google Scholar 

  34. Xu P.L.: A hybrid global optimization method: the multi-dimensional case. J. Comput. Appl. Math. 155, 423–446 (2003)

    Google Scholar 

  35. Hedar A.R., Fukushima M.: Hybrid simulated annealing and direct search method for nonlinear unconstrained global optimization. Optim. Methods Softw. 17, 891–912 (2002)

    Article  Google Scholar 

  36. Barhen J., Protopopescu V., Reister D.: TRUST: a deterministic algorithm for global optimization. Science 276, 1094–1097 (1997)

    Article  Google Scholar 

  37. Basso P.: Iterative methods for the localization of the global maximum. SIAM J. Numer. Anal. 19, 781–792 (1982)

    Article  Google Scholar 

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

    Article  Google Scholar 

  39. Floudas C.: Global Optimization: Theory, Methods and Applications. Kluwer, Dordrecht (2000)

    Google Scholar 

  40. Hansen E.: Global optimization using interval-analysis—the multidimensional case. Numer. Math. 34, 247–270 (1980)

    Article  Google Scholar 

  41. Hansen E.R.: Global optimization using interval analysis— one-dimensional case. J. Optim. Theory Appl. 29, 331–344 (1979)

    Article  Google Scholar 

  42. Hansen E.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992)

    Google Scholar 

  43. Ichida K., Fujii Y.: Interval arithmetic method for global optimization. Computing 23, 85–97 (1979)

    Article  Google Scholar 

  44. Kearfott R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht (1996)

    Google Scholar 

  45. Ratschek H., Rokne J.: New Computer Methods for Global Optimization. Ellis Horwood, Chichester (1988)

    Google Scholar 

  46. Tarvainen M., Tiira T., Husebye E.S.: Locating regional seismic events with global optimization based on interval arithmetic. Geophys. J. Int. 138, 879–885 (1999)

    Article  Google Scholar 

  47. Land A.H., Doig A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)

    Article  Google Scholar 

  48. Clausen, J.: Branch and bound algorithms—principles and examples. In: Department of Computer Science, University of Copenhagen (1999, March)

  49. Tawarmalani M., Sahinidis N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer, Boston (2002)

    Google Scholar 

  50. Khajavirad, A., Michalek, J.J.: A deterministic Lagrangian-based global optimization approach for quasiseparable nonconvex mixed-integer nonlinear programs. J. Mech. Design 131, 051009 (8pp)

  51. Qu S.J., Ji Y., Zhang K.C.: A deterministic global optimization algorithm based on a linearizing method for nonconvex quadratically constrained programs. Math. Comput. Model. 48, 1737–1743 (2008)

    Article  Google Scholar 

  52. Jiao H.W., Chen Y.Q.: A note on a deterministic global optimization algorithm. Appl. Math. Comput. 202, 67–70 (2008)

    Article  Google Scholar 

  53. Wu Y., Lai K.K., Liu Y.J.: Deterministic global optimization approach to steady-state distribution gas pipeline networks. Optim. Eng. 8, 259–275 (2007)

    Article  Google Scholar 

  54. Long C.E., Polisetty P.K., Gatzke E.P.: Deterministic global optimization for nonlinear model predictive control of hybrid dynamic systems. Int. J. Robust Nonlinear Control 17, 1232–1250 (2007)

    Article  Google Scholar 

  55. Lin Y.D., Stadtherr M.A.: Deterministic global optimization of nonlinear dynamic systems. AICHE J. 53, 866–875 (2007)

    Article  Google Scholar 

  56. Ji Y., Zhang K.C., Qu S.H.: A deterministic global optimization algorithm. Appl. Math. Comput. 185, 382–387 (2007)

    Article  Google Scholar 

  57. Lin Y., Stadtherr M.A.: Deterministic global optimization for parameter estimation of dynamic systems. Ind Eng Chem Res 45, 8438–8448 (2006)

    Article  Google Scholar 

  58. Long C.E., Polisetty P.K., Gatzke E.P.: Nonlinear model predictive control using deterministic global optimization. J. Process Control 16, 635–643 (2006)

    Article  Google Scholar 

  59. Lin Y.D., Stadtherr M.A.: Deterministic global optimization of molecular structures using interval analysis. J. Comput. Chem. 26, 1413–1420 (2005)

    Article  Google Scholar 

  60. Sun, W.T., Shu, J.W., Zheng, W.M.: Deterministic global optimization with a neighbourhood determination algorithm based on neural networks. In: Advances in Neural Networks—ISNN 2005, Pt 1, Proceedings, vol. 3496, pp. 700–705 (2005)

  61. Messine F.: Deterministic global optimization using interval constraint propagation techniques. Rairo Oper. Res. 38, 277–293 (2004)

    Article  Google Scholar 

  62. Adjiman C.S., Papamichail I.: A deterministic global optimization algorithm for problems with nonlinear dynamics. Front. Glob. Optim. 74, 1–23 (2004)

    Google Scholar 

  63. Gau C.Y.T., Schrage L.E.: Implementation and testing of a branch-and-bound based method for deterministic global optimization: operations research applications. Front. Glob. Optim. 74, 145–164 (2004)

    Google Scholar 

  64. Lin Y., Stadtherr M.A.: Advances in interval methods for deterministic global optimization in chemical engineering. J. Glob. Optim. 29, 281–296 (2004)

    Article  Google Scholar 

  65. Bartholomew-Biggs M.C., Parkhurst S.C., Wilson S.R.: Global optimization—stochastic or deterministic?. Stoch. Algorithms Found. Appl. 2827, 125–137 (2003)

    Article  Google Scholar 

  66. Sambridge M.: Geophysical inversion with a neighbourhood algorithm—II. Appraising the ensemble. Geophys. J. Int. 138, 727–746 (1999)

    Article  Google Scholar 

  67. Sambridge M.: Geophysical inversion with a neighbourhood algorithm—I. Searching a parameter space. Geophys. J. Int. 138, 479–494 (1999)

    Article  Google Scholar 

  68. Sambridge M., Braun J., Mcqueen H.: Geophysical parametrization and interpolation of irregular data using natural neighbors. Geophys. J. Int. 122, 837–857 (1995)

    Article  Google Scholar 

  69. Locatelli M., Wood G.R.: Objective function features providing barriers to rapid global optimization. J. Glob. Optim. 31, 549–565 (2005)

    Article  Google Scholar 

  70. Locatelli M.: On the multilevel structure of global optimization problems. Comput. Optim. Appl. 30, 5–22 (2005)

    Article  Google Scholar 

  71. Daubechies I., Mallat S., Willsky A.S.: Special issue on wavelet transforms and multiresolution signal analysis—introduction. IEEE Trans. Inf. Theory 38, 529–531 (1992)

    Google Scholar 

  72. Daubechies I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36, 961–1005 (1990)

    Article  Google Scholar 

  73. Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Google Scholar 

  74. Mallat S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)

    Article  Google Scholar 

  75. Meyer, Y.: Principle d’incertitude, basis Hilbertiennes et algebras d’operateurs. In: Bourbaki Seminar (1885–1986)

  76. Daubechies I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)

    Article  Google Scholar 

  77. Daubechies I., Paul T.: Time frequency localization operators—a geometric phase-space approach 2. The use of dilations. Inverse Probl. 4, 661–680 (1988)

    Article  Google Scholar 

  78. Kalantari B., Rosen J.B.: Construction of large-scale global minimum concave quadratic test problems. J. Optim. Theory Appl. 48, 303–313 (1986)

    Article  Google Scholar 

  79. Floudas C., Pardalos P.M.: A collection of test problems for constrained global optimization algorithms. In: Goos GaH, J. Lecture Notes in Computer Science, Springer, Berlin (1990)

  80. Khoury B.N., Pardalos P.M., Du D.Z.: A test problem generator for the Steiner problem in graphs. ACM Trans. Math. Softw. 19, 509–522 (1993)

    Article  Google Scholar 

  81. Schoen F.: A wide class of test functions for global optimization. J. Glob. Optim. 3, 133–137 (1993)

    Article  Google Scholar 

  82. Mathar R., Zilinskas A.: A class of test functions for global optimization. J. Glob. Optim. 5, 195–199 (1994)

    Article  Google Scholar 

  83. Facchinei F., Judice J., Soares J.: Generating box-constrained optimization problems. ACM Trans. Math. Softw. 23, 443–447 (1997)

    Article  Google Scholar 

  84. Gaviano R., Lera D.: Test functions with variable attraction regions for global optimization problems. J. Glob. Optim. 13, 207–223 (1998)

    Article  Google Scholar 

  85. Gaviano M., Kvasov D.E., Lera D., Sergeyev Y.D.: Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29, 469–480 (2003)

    Article  Google Scholar 

  86. Mishra, S.: Some new test functions for global optimization and performance of repulsive particle swarm method. In: MPRA (2006)

  87. Addis B., Locatelli M.: A new class of test functions for global optimization. J. Glob. Optim. 38, 479–501 (2007)

    Article  Google Scholar 

  88. Jones D.R., Perttunen C.D., Stuckman B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)

    Article  Google Scholar 

  89. Liang, J.J., Suganthan, P.N., Deb, K.: Novel composition test functions for numerical global optimization. In: 2005 IEEE Swarm Intelligence Symposium, Pasadena, pp. 68–75. IEEE Press (2005)

  90. Schwefel H.-P.: Numerical Optimization of Computer Models. Wiley, New York (1981)

    Google Scholar 

  91. Ackley D.H.: A Connectionist Machine for Genetic Hillclimbing. Springer, Boston (1987)

    Google Scholar 

  92. Conn A.R., Gould N.I.M., Toint P.L.: Testing a class of methods for solving minimization problems with simple bounds on the variables. Math. Comput. 50, 399–430 (1988)

    Article  Google Scholar 

  93. Branch M.A., Coleman T.F., Li Y.Y.: A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems. SIAM J. Sci. Comput. 21, 1–23 (1999)

    Article  Google Scholar 

  94. Dixon L.C.W., Szego G.P.: The optimization problem: an introduction. In: Dixon, L.C.W., Szego, G.P. (eds) Towards Global Optimization II, North Holland, New York (1978)

    Google Scholar 

  95. Goldstei A.A., Price J.F.: Descent from local minima. Math. Comput. 25, 569–574 (1971)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ZHOU PEI-YUAN Center for Applied Mathematics, Tsinghua University, 100084, Beijing, China

    Weitao Sun

  2. Department of Computer Science and Technology, Tsinghua University, 100084, Beijing, China

    Yuan Dong

Authors
  1. Weitao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weitao Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, W., Dong, Y. Study of multiscale global optimization based on parameter space partition. J Glob Optim 49, 149–172 (2011). https://doi.org/10.1007/s10898-010-9540-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-010-9540-x

Keywords

Search

Navigation