Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint

Hoai An Le Thi; Mohand Ouanes

RAIRO - Operations Research (2006)

  • Volume: 40, Issue: 3, page 285-302
  • ISSN: 0399-0559

Abstract

top
The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems with both twice differentiable function and constraint, we can propose an efficient algorithm based on Branch and Bound techniques. The method is first displayed in the simple case with an interval constraint. The extension is displayed afterwards to the general case with an additional nonconvex twice differentiable constraint. A quadratic bounding function which is better than the well known linear underestimator is proposed while w-subdivision is added to support the branching procedure. Computational results on several and various types of functions show the efficiency of our algorithms and their superiority with respect to the existing methods.

How to cite

top

Le Thi, Hoai An, and Ouanes, Mohand. "Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint." RAIRO - Operations Research 40.3 (2006): 285-302. <http://eudml.org/doc/249771>.

@article{LeThi2006,
abstract = { The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems with both twice differentiable function and constraint, we can propose an efficient algorithm based on Branch and Bound techniques. The method is first displayed in the simple case with an interval constraint. The extension is displayed afterwards to the general case with an additional nonconvex twice differentiable constraint. A quadratic bounding function which is better than the well known linear underestimator is proposed while w-subdivision is added to support the branching procedure. Computational results on several and various types of functions show the efficiency of our algorithms and their superiority with respect to the existing methods. },
author = {Le Thi, Hoai An, Ouanes, Mohand},
journal = {RAIRO - Operations Research},
keywords = {Global optimization; branch and bound; quadratic underestimation; w-subdivision.; global optimization; quadratic underestimation; -subdivision},
language = {eng},
month = {11},
number = {3},
pages = {285-302},
publisher = {EDP Sciences},
title = {Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint},
url = {http://eudml.org/doc/249771},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Le Thi, Hoai An
AU - Ouanes, Mohand
TI - Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint
JO - RAIRO - Operations Research
DA - 2006/11//
PB - EDP Sciences
VL - 40
IS - 3
SP - 285
EP - 302
AB - The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems with both twice differentiable function and constraint, we can propose an efficient algorithm based on Branch and Bound techniques. The method is first displayed in the simple case with an interval constraint. The extension is displayed afterwards to the general case with an additional nonconvex twice differentiable constraint. A quadratic bounding function which is better than the well known linear underestimator is proposed while w-subdivision is added to support the branching procedure. Computational results on several and various types of functions show the efficiency of our algorithms and their superiority with respect to the existing methods.
LA - eng
KW - Global optimization; branch and bound; quadratic underestimation; w-subdivision.; global optimization; quadratic underestimation; -subdivision
UR - http://eudml.org/doc/249771
ER -

References

top
  1. W. Baritompa and A. Cutler, Accelerations for global optimization covering methods using second derivates. J. Global Optim.4 (1994) 329–341.  
  2. E. Baumann, Optimal centered forms. BIT28 (1988) 80–87.  
  3. C de Boor, A practical Guide to Splines Applied Mathematical Sciences. Springer-Verlag (1978).  
  4. J. Calvin, and A. Ilinskas, On the convergence of the P-algorithm for one-dimensional global optimization of smooth functions. J. Optim. Theory Appl.102 (1999) 479–495.  
  5. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies Math. Appl. (1979).  
  6. J.E. Falk and R.M. Soland, An algorithm for separable nonconvex programming problems. Manage. Sci.15 (1969) 550–569.  
  7. D. Famularo, Y.D. Sergeyev and P. Pugliese, Test Problems for Lipschitz Univariate Global Optimization with Multiextremal Constraints. Publication online.  
  8. C. Floudas and V. Visweswaran, A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs-I. Theory. Comput. Chemical Engin.14 (1990) 1394–1417.  
  9. C.A. Floudas, P.M. Pardalos, C.S. Adjiman, W.R. Esposito, Z.H. Gümüs, S.T. Harding, J.L. Klepeis, C.A. Meyer and C.A. Schweiger, Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers (1999).  
  10. V.P. Gergel, A Global Optimization Algorithm for Multivariate functions with Lipschitzian First Derivatives, J. Global Optim. 10 (1997) 257–281.  
  11. K. Glashoff and S.A. Gustafson, Linear Optimization and Approximation. Amplitude Math. Sciences, Springer-Verlag (1983).  
  12. E. Gourdin, B. Jaumard and R. Ellaia, Global Optimization of Hölder Functions. J. Global Optim.8 (1996) 323–348.  
  13. E. Hansen, Global optimization using interval analysis the multi-dimensional case. Numer. Math.34 (1980) 247270.  
  14. E. Hansen, Global Optimization using Interval Analysis. Pure Appl. Math.165, Marcel Dekker, New York (1992).  
  15. P. Hansen, B. Jaumard and J. Xiong, Decomposition and interval arithmetic applied to minimization of polynomial and rational functions. J. Global Optim.3 (1993) 421–437.  
  16. P. Hansen, B. Jaumard and J. Xiong, Cord-slope form of Taylor's expansion in univariate global optimization. J. Optim. Theory Appl.80 (1994) 441–464.  
  17. P. Hansen, B. Jaumard and S.-H. Lu, Global Optimization of Univariate Functions by Sequential Polynomial Approximation. Int. J. Comput. Math.28 (1989) 183–193.  
  18. P. Hansen, B. Jaumard and S.-H. Lu, Global Optimization of Univariate Lipschitz Functions: 2. New Algorithms and Computational Comparison. Math. Program.55 (1992) 273–292.  
  19. R. Horst and P.M. Pardalos, Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (1995).  
  20. R. Horst and H. Tuy, Global Optimization – Deterministic Approaches. Springer-Verlag, Berlin (1993).  
  21. H.A. Le Thi and T. Pham Dinh, A Branch-and-Bound method via D.C. Optimization Algorithm and Ellipsoidal technique for Box Constrained Nonconvex Quadratic Programming Problems. J. Global Optim.13 (1998) 171–206.  
  22. M. Locatelli and F. Schoen, An adaptive stochastic global optimization algorithm for one dimensional functions. Ann. Oper. Res.58 (1995) 263–278.  
  23. F. Messine and J. Lagouanelle, Enclosure methods for multivariate differentiable functions and application to global optimization. J. Universal Comput. Sci.4 (1998) 589–603.  
  24. R. Moore, Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, (1966).  
  25. P.M. Pardalos and H.E. Romeijn, Handbook of Global Optimization, Volume 2. Nonconvex Optimization and Its Applications, Springer, Boston/Dordrecht/London (2002).  
  26. S.A. Pijavskii, An Algorithm for Finding the Absolute Extremum of a Function. USSR. Comput. Math. Math. Physics12 (1972) 57–67.  
  27. T. Quynh Phong, L.T. Hoai An and P. Dinh Tao, On the global solution of linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method. RAIRO: Oper. res.30 (1996) 31–49.  
  28. H. Ratschek and J. Rokne, New Computer Methods for Global Optimization. Wiley, New York (1982).  
  29. D. Ratz, A nonsmooth global optimization technique using slopes the one-dimensional case. J. Global Optim.14 (1999) 365–393.  
  30. A.H.G. Rinnooy and G.H. Timmer, Global Optimization: A Survey, in Handbooks of Operations Research, edited by G.L. Nemhauser et al. North-Holland, Elsevier Science Publishers (1989) 631–662.  
  31. Ya.D. Sergeyev and V.A. Grishagin, A Parallel Method for Finding the Global Minimum of Univariate Functions. J. Optim. Theory Appl.80 (1994) 513–536.  
  32. Ya.D. Sergeyev, An One-Dimensional Deterministic Global Minimization Algorithm. Russian J. Comput. Math. Math. Phys.35 705–717.  
  33. A. Törn and A. Zilinskas, Global Optimization, edited by G. Goos and J. Hartmanis. Springer, Berlin, Lect. Notes Comput. Sci.350 (1989).  
  34. V. Visweswaran and C.A. Floudas, Global Optimization of Problems with Polynomial Functions in One Variable, in Recent Advances in Global Optimization, edited by A. Floudas and P.M. Pardalos. Princeton, Princeton University Press, pp. 165–199.  
  35. Ya.D. Sergeyev, Global one-dimensional optimization using smooth auxiliary functions. Math. Program.81 (1998) 127–146.  
  36. X. Wang and T.-S. Chang, An Improved Univariate Global Optimization Algorithm with Improved Linear Bounding Functions. J. Global Optim.8 (1996) 393–411.  
  37. A. Zilinskas, On Global One-Dimensional Optimization. Engineering Cybernetics, Izvestija AN USSR4 (1976) 71–74.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.