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 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.
@article{RO_2006__40_3_285_0,
author = {Le Thi, Hoai An and Ouanes, Mohand},
title = {Convex quadratic underestimation and {Branch} and {Bound} for univariate global optimization with one nonconvex constraint},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {285--302},
publisher = {EDP-Sciences},
volume = {40},
number = {3},
year = {2006},
doi = {10.1051/ro:2006024},
mrnumber = {2276160},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ro:2006024/}
}
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 - Recherche Opérationnelle PY - 2006 SP - 285 EP - 302 VL - 40 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2006024/ DO - 10.1051/ro:2006024 LA - en ID - RO_2006__40_3_285_0 ER -
%0 Journal Article %A Le Thi, Hoai An %A Ouanes, Mohand %T Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint %J RAIRO - Operations Research - Recherche Opérationnelle %D 2006 %P 285-302 %V 40 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro:2006024/ %R 10.1051/ro:2006024 %G en %F RO_2006__40_3_285_0
Le Thi, Hoai An; Ouanes, Mohand. Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint. RAIRO - Operations Research - Recherche Opérationnelle, Tome 40 (2006) no. 3, pp. 285-302. doi : 10.1051/ro:2006024. http://archive.numdam.org/articles/10.1051/ro:2006024/
[1] and, Accelerations for global optimization covering methods using second derivates. J. Global Optim. 4 (1994) 329-341. | Zbl
[2] , Optimal centered forms. BIT 28 (1988) 80-87. | Zbl
[3] C de Boor, A practical Guide to Splines Applied Mathematical Sciences. Springer-Verlag (1978). | MR | Zbl
[4] , and, On the convergence of the P-algorithm for one-dimensional global optimization of smooth functions. J. Optim. Theory Appl. 102 (1999) 479-495. | Zbl
[5] , The Finite Element Method for Elliptic Problems. Studies Math. Appl. (1979). | MR | Zbl
[6] and, An algorithm for separable nonconvex programming problems. Manage. Sci. 15 (1969) 550-569. | Zbl
[7] , and, Test Problems for Lipschitz Univariate Global Optimization with Multiextremal Constraints. Publication online. | MR | Zbl
[8] and, A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs-I. Theory. Comput. Chemical Engin. 14 (1990) 1394-1417.
[9] ,,,,,,, and, Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers (1999). | MR | Zbl
[10] , A Global Optimization Algorithm for Multivariate functions with Lipschitzian First Derivatives, J. Global Optim. 10 (1997) 257-281. | Zbl
[11] and, Linear Optimization and Approximation. Amplitude Math. Sciences, Springer-Verlag (1983). | MR | Zbl
[12] , and, Global Optimization of Hölder Functions. J. Global Optim. 8 (1996) 323-348. | Zbl
[13] , Global optimization using interval analysis the multi-dimensional case. Numer. Math. 34 (1980) 247270. | MR | Zbl
[14] , Global Optimization using Interval Analysis. Pure Appl. Math. 165, Marcel Dekker, New York (1992). | MR | Zbl
[15] , and, Decomposition and interval arithmetic applied to minimization of polynomial and rational functions. J. Global Optim. 3 (1993) 421-437. | Zbl
[16] , and, Cord-slope form of Taylor's expansion in univariate global optimization. J. Optim. Theory Appl. 80 (1994) 441-464. | Zbl
[17] , and, Global Optimization of Univariate Functions by Sequential Polynomial Approximation. Int. J. Comput. Math. 28 (1989) 183-193. | Zbl
[18] , and, Global Optimization of Univariate Lipschitz Functions: 2. New Algorithms and Computational Comparison. Math. Program. 55 (1992) 273-292. | Zbl
[19] and, Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (1995). | MR | Zbl
[20] and, Global Optimization - Deterministic Approaches. Springer-Verlag, Berlin (1993). | Zbl
[21] and, 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. | Zbl
[22] and, An adaptive stochastic global optimization algorithm for one dimensional functions. Ann. Oper. Res. 58 (1995) 263-278. | Zbl
[23] and, Enclosure methods for multivariate differentiable functions and application to global optimization. J. Universal Comput. Sci. 4 (1998) 589-603. | Zbl
[24] , Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, (1966). | MR | Zbl
[25] and, Handbook of Global Optimization, Volume 2. Nonconvex Optimization and Its Applications, Springer, Boston/Dordrecht/London (2002). | Zbl
[26] , An Algorithm for Finding the Absolute Extremum of a Function. USSR. Comput. Math. Math. Physics 12 (1972) 57-67. | Zbl
[27] , and, On the global solution of linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method. RAIRO: Oper. res. 30 (1996) 31-49. | Numdam | Zbl
[28] and, New Computer Methods for Global Optimization. Wiley, New York (1982). | Zbl
[29] , A nonsmooth global optimization technique using slopes the one-dimensional case. J. Global Optim. 14 (1999) 365-393. | Zbl
[30] and, Global Optimization: A Survey, in Handbooks of Operations Research, edited by G.L. Nemhauser et al. North-Holland, Elsevier Science Publishers (1989) 631-662. | Zbl
[31] and, A Parallel Method for Finding the Global Minimum of Univariate Functions. J. Optim. Theory Appl. 80 (1994) 513-536. | Zbl
[32] , An One-Dimensional Deterministic Global Minimization Algorithm. Russian J. Comput. Math. Math. Phys. 35 705-717. | Zbl
[33] and, Global Optimization, edited by G. Goos and J. Hartmanis. Springer, Berlin, Lect. Notes Comput. Sci. 350 (1989). | MR | Zbl
[34] and, 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] , Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81 (1998) 127-146. | Zbl
[36] and, An Improved Univariate Global Optimization Algorithm with Improved Linear Bounding Functions. J. Global Optim. 8 (1996) 393-411. | Zbl
[37] , On Global One-Dimensional Optimization. Engineering Cybernetics, Izvestija AN USSR 4 (1976) 71-74. | Zbl
Cité par Sources :
