Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-17T15:25:55.131Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  08 November 2006

Hoai An Le Thi  and
Mohand Ouanes
Hoai An Le Thi
Affiliation:
Laboratoire de l'Informatique Théorique et Appliquée, UFR Scientifique MIM Université Paul Verlaine – Metz, Ile du Saulcy, 57045 Metz, France; lethi@univ-metz.fr
Mohand Ouanes
Affiliation:
Laboratoire de l'Informatique Théorique et Appliquée, UFR Scientifique MIM Université Paul Verlaine – Metz, Ile du Saulcy, 57045 Metz, France; lethi@univ-metz.fr Département de Mathématiques, Faculté des Sciences, Université de Tizi-Ouzou, Algeria.
Get access

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.

Keywords

Global optimizationbranch and boundquadratic underestimationw-subdivision.
Type
Research Article
Information
RAIRO - Operations Research , Volume 40 , Issue 3 , July 2006 , pp. 285 - 302
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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