Geodesic
Multidimensional global optimization using the first derivatives
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 5, pp. 743-752 Cet article a éte moissonné depuis la source Math-Net.Ru

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We propose a new multidimensional algorithm for solving global optimization problems with the objective function having Lipschitzean first derivatives and determined over a multidimensional interval. The method does not belong to the class of multistart algorithms. It is based on the following three new proposals and is an illustration how it is possible to generalize to the multidimensional case the one-dimensional algorithms belonging to the Classes of adaptive partition and characteristical global optimization methods. The first proposal is to estimate the local Lipschitz constants for derivatives in different subintervals of the search region during the course of optimization to provide a local tuning on the behavior of the objective function. The second one is a new partitioning scheme providing an efficient keeping of the search information. The last proposal is a way to calculate characteristics of multidimensional intervals to provide convergence to the global minimizers. 
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Ya. D. Sergeyev. Multidimensional global optimization using the first derivatives. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 5, pp. 743-752. http://geodesic.mathdoc.fr/item/ZVMMF_1999_39_5_a3/

[1] Archetti F., Schoen F., “A survey on the global optimization problems: General theory and computational approaches”, Ann. Operat. Res., 1 (1984), 87–110 | DOI | Zbl

[2] Butz A. R., “Space filling curves and mathematical programming”, Inform. Control, 12:4 (1968), 314–330 | DOI | MR | Zbl

[3] Dixon L. C. W., Szegő G. P., Towards global optimization, v. 2, North-Holland, Amsterdam, 1978 | MR | Zbl

[4] Evtushenko Yu. G., “Numerical methods for finding global extrema of a non-uniform mesh”, USSR Comput. Maths. Math. Phys., 11:6 (1971), 38–54 | DOI

[5] Galperin E. A., “The cubic algorithm”, J. Math. Analys. and Appl., 112:2 (1985), 635–640 | DOI | MR | Zbl

[6] Hansen P., Jaumard B., “Lipschitz optimization”, Handbook of Global Optimization, Kluwer Acad. Publs, Dordrecht, 1995, 407–493 | MR | Zbl

[7] Horst R., Pardalos P. M. (eds.), Handbook of global optimization, Kluwer Acad. Publs, Dordrecht, 1995 | MR

[8] Horst R., Tuy H., Global optimization – deterministic approaches, 3d ed., Springer Verlag, Berlin, 1996 | MR

[9] Korotkich V. V., “On a mechanism of natural formation and its use in global optimization”, Proc. 3d Workshop on Global Optimizat. (Szeged, 1995), 57–59

[10] Mladineo R. H., “An algorithm for finding the global maximum of a multimodal, multivariate function”, Math. Program., 34 (1986), 188–200 | DOI | MR | Zbl

[11] P. M. Pardalos, J. B. Rosen (eds.), Computational methods in global optimization, Ann. Operat. Res., 25, 1990 | Zbl

[12] Piyavskii S. A., “An algorithm for finding the absolute extremum of a function”, USSR Comput. Maths. Math. Phys., 12:4 (1972), 57–67 | DOI

[13] Ratschek H., Rokne J., New computer methods for global optimization, Ellis Horwood, Chichester, 1988 | MR | Zbl

[14] Optimization, North-Holland, Amsterdam, 1989 | MR

[15] Sergeyev Ya. D., “Divide the best” global optimization algorithms, Rept. No 2/94, Univ. Calabria, Dept. Math., 1994

[16] Sergeyev Ya. D., A global optimization algorithm using derivatives and local tuning, Rept. No 1, ISI-CNR, 1994

[17] Sergeyev Ya. D., “A method using local tuning for minimizing functions with Lipschitz derivatives”, Developments in Global Optimizat., Kluwer Acad. Publs, Dordrecht, 1997, 199–215 | MR

[18] Sergeyev Ya. D., “An information global optimization algorithm with local tuning”, SIAM J. Optimizat., 5:4 (1995), 858–870 | DOI | MR | Zbl

[19] Sergeyev Ya. D., “A one-dimensional deterministic global minimization algorithm”, Comput. Maths. Math. Phys., 35:5 (1995), 553–562 | MR

[20] Sergeyev Ya. D., Grishagin V. A., “Sequential and parallel algorithms for global optimization”, Optimizat. Methods and Software, 3 (1994), 111–124 | DOI

[21] Sergeyev Ya. D., Grishagin V. A., “A parallel algorithm for finding the global minimum of univariate functions”, J. Optimizat. Theory and Appl., 80:3 (1994), 513–536 | DOI | MR | Zbl

[22] Sergeyev Ya. D., Markin D. L., “An algorithm for solving global optimization problems with nonlinear constraints”, J. Global Optimizat., 7 (1995), 407–419 | DOI | MR | Zbl

[23] Strongin R. G., Numerical methods in multiextremal problems, Nauka, Moscow, 1978 | MR | Zbl

[24] Törn A., Žilinskas A., Global optimization, Lect. Notes in Comput. Sci., 350, Springer-Verlag, Berlin, 1989 | MR

[25] Wood G. R., “Multidimensional bisection and global optimization”, Comput. and Math. with Applic., 21 (1991), 161–172 | DOI | MR | Zbl

[26] Zhigljavsky A. A., Theory of global random search, Kluwer Acad. Publs, Dordrecht, 1991 | MR

[27] Pinter J., “Convergence qualification of adaptive partition algorithms in global optimization”, Math. Program., 56 (1992), 343–360 | DOI | MR | Zbl

[28] Grishagin V. A., “On convergence conditions for a class of global optimization algorithms”, Numer. Methods for Nonlinear Programming, Kharkiv State Univ., Kharkiv, 1979, 82–84

[29] Grishagin V. A., Sergeyev Ya. D., Strongin R. G., “Parallel characteristical global optimization algorithms”, J. Global Optimizat., 10 (1997), 185–206 | DOI | MR | Zbl

[30] Breiman L., Cutler A., “A deterministic algorithm for global optimization”, Math. Program., 58 (1993), 179–199 | DOI | MR | Zbl

[31] Baritompa W., “Accelerations for a variety of global optimization methods”, J. Global Optimizat., 4:1 (1994), 37–45 | DOI | MR | Zbl

[32] Gergel V. P., “A global search algorithm using derivatives”, Systems Dynamics and Optimizat., N. Novgorod Univ. Press, N. Novgorod, 1992, 161–178 | MR | Zbl

[33] Pinter J., “Extended univariate algorithms forn-dimensional global optimization”, Computing, 36 (1986), 91–103 | DOI | MR | Zbl

[34] Meewella C. C., Mayne D. Q., “An algorithm for global optimization of Lipschitz continuous functions”, J. Optimizat. Theory and Appl., 57:2 (1988), 307–322 | DOI | MR | Zbl

[35] Meewella C. C., Mayne D. Q., “Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions”, J. Optimizat. Theory and Appl., 61:2 (1989), 247–270 | DOI | MR | Zbl

[36] Baritompa W., Cutler A., “Accelerations for global optimization covering methods using second derivatives”, J. Global Optimizat., 4:3 (1994), 329–341 | DOI | MR | Zbl