zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A hybrid global optimization method: The multi-dimensional case. (English) Zbl 1020.65033
Summary: We extend the hybrid global optimization method proposed by {\it P. Xu} [J. Comput. Appl. Math. 147, 301-314 (2002; Zbl 1013.65063)] for the one-dimensional case to the multi-dimensional case. The method consists of two basic components: local optimizers and feasible point finders. Local optimizers guarantee efficiency and speed of producing a local optimal solution in the neighbourhood of a feasible point. Feasible point finders provide the theoretical guarantee for the new method to always produce the global optimal solution(s) correctly. If a nonlinear nonconvex inverse problem has multiple global optimal solutions, our algorithm is capable of finding all of them correctly. Three synthetic examples, which have failed simulated annealing and genetic algorithms, are used to demonstrate the proposed method.

MSC:
65K05Mathematical programming (numerical methods)
90C30Nonlinear programming
90C53Methods of quasi-Newton type
Software:
INTOPT_90
WorldCat.org
Full Text: DOI
References:
[1] Aarts, E.; Korst, J.: Simulated annealing and Boltzmann machines--A stochastic approach to combinatorial optimization and neural computing. (1997) · Zbl 0674.90059
[2] Barhen, J.; Protopopescu, V.; Reister, D.: Trusta deterministic algorithm for global optimization. Science 276, 1094-1097 (1997) · Zbl 1226.90073
[3] Basso, P.: Iterative methods for the localization of the global maximum. SIAM J. Numer. anal. 19, 781-792 (1982) · Zbl 0483.65038
[4] Basu, A.; Frazer, L. N.: Rapid determination of the critical temperature in simulated annealing inversion. Science 249, 1409-1412 (1990) · Zbl 1226.86005
[5] Bertsekas, D. P.: Constrained optimization and Lagrange multiplier methods. (1982) · Zbl 0572.90067
[6] Caprani, O.; Godthaab, B.; Madsen, K.: Use of a real-valued local minimum in parallel interval global optimization. Interval comput. 2, 71-82 (1993) · Zbl 0829.65081
[7] Cary, P. W.; Chapman, C. H.: Automatic 1-D waveform inversion of marine seismic reflection data. Geophys. J. 93, 527-546 (1988) · Zbl 0637.73028
[8] Chunduru, R. K.; Sen, M. K.; Stoffa, P. L.: Hybrid optimization methods for geophysical inversion. Geophysics 62, 1196-1207 (1997)
[9] Cohn, H.; Fielding, M.: Simulated annealingsearching for an optimal temperature schedule. SIAM J. Optim. 9, 779-802 (1999) · Zbl 0957.60072
[10] Daniel, J. W.: Newton’s method for nonlinear inequalities. Numer. math. 21, 381-387 (1973) · Zbl 0293.65039
[11] Dennis, J. E.; El-Alem, M.; Williamson, K.: A trust-region approach to nonlinear systems of equalities and inequalities. SIAM J. Optim. 9, 291-315 (1999) · Zbl 0957.65058
[12] Dennis, J. E.; Schnabel, R. B.: Numerical methods for unconstrained optimization and nonlinear equations. (1996) · Zbl 0847.65038
[13] Fallat, M. R.; Dosso, S. E.: Geoacoustic inversion via local, global, and hybrid algorithms. J. acoust. Soc. amer. 105, 3219-3230 (1999)
[14] Floudas, C. A.: Deterministic global optimization: theory, methods and applications. (2000)
[15] Forrest, S.: Genetic algorithmsprinciples of natural selection applied to computation. Science 261, 872-878 (1993)
[16] Gerstoft, P.: Inversion of acoustic data using a combination of genetic algorithms and the Gauss--Newton approach. J. acoust. Soc. amer. 97, 2181-2190 (1995)
[17] Goldberg, D. E.: Genetic algorithms in search, optimization and machine learning. (1989) · Zbl 0721.68056
[18] Hansen, E.: Global optimization using interval analysisthe one-dimensional case. J. optim. Theor. appl. 29, 331-344 (1979) · Zbl 0388.65023
[19] Hansen, E.: Global optimization using interval analysisthe multi-dimensional case. Numer. math. 34, 247-270 (1980) · Zbl 0442.65052
[20] Hansen, E.: Global optimization using interval analysis. (1992) · Zbl 0762.90069
[21] Hansen, E.; Sengupta, S.: Global constrained optimization using interval analysis. Interval mathematics, 25-47 (1980) · Zbl 0537.65052
[22] Holland, J. H.: Adaptation in natural and artificial systems. (1975) · Zbl 0317.68006
[23] Hong, H.: Heuristic search and pruning in polynomial constraints satisfaction. Ann. math. Artif. intellig. 19, 319-334 (1997) · Zbl 0880.68069
[24] Hopfield, J. J.; Tank, D. W.: Computing with neural circuitsa model. Science 233, 625-633 (1986)
[25] Ichida, K.; Fujii, Y.: An interval arithmetic method for global optimization. Computing 23, 85-97 (1979) · Zbl 0403.65028
[26] Kearfott, R. B.: Rigorous global search: continuous problems. (1996) · Zbl 0876.90082
[27] Kirkpatrick, S.; Gelatt, C. D.; Jr., M. P. Vecchi: Optimization by simulated annealing. Science 220, 671-680 (1983) · Zbl 1225.90162
[28] Liu, P.; Hartzell, S.; Stephenson, W.: Non-linear multiparameter inversion using a hybrid global search algorithmapplications in reflection seismology. Geophys. J. Int. 122, 991-1000 (1995)
[29] Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller, E.: Equation of state calculations by fast computing machines. J. chem. Phys. 21, 1087-1092 (1953)
[30] Moore, R. E.: Interval analysis. (1966) · Zbl 0176.13301
[31] Nelder, J. A.; Mead, R.: A simplex method for function minimization. Comput. J. 7, 308-313 (1965) · Zbl 0229.65053
[32] Neumaier, A.: The enclosure of solutions of parameter-dependent systems of equations. Reliability in computing, 269-286 (1988) · Zbl 0651.65044
[33] Neumaier, A.: Interval methods for systems of equations. (1990) · Zbl 0715.65030
[34] Nocedal, J.; Wright, S. J.: Numerical optimization. (1999) · Zbl 0930.65067
[35] Parker, R. L.: Geophysical inverse theory. (1994) · Zbl 0812.35159
[36] Piyavskii, S. A.: An algorithm for finding the absolute extremum of a function. USSR comput. Math. math. Phys. 12, 57-67 (1972)
[37] Polyak, B. T.: Gradient methods for solving equations and inequalities. USSR comput. Math. math. Phys. 4, 17-32 (1964)
[38] Pshenichnyi, B. N.: Newton’s method for the solution of systems of equalities and inequalities. Math. notes acad. Sci. USSR 8, 827-830 (1970)
[39] Ratschek, H.; Rokne, J.: New computer methods for global optimization. (1988) · Zbl 0648.65049
[40] Robinson, S. M.: Extension of Newton’s method to nonlinear functions with values in a cone. Numer. math. 19, 341-347 (1972) · Zbl 0227.65040
[41] Scales, J. A.; Smith, M. L.; Fischer, T. L.: Global optimization methods for multimodal inverse problems. J. comput. Phys. 103, 258-268 (1992) · Zbl 0765.65062
[42] Seber, G. A. F.; Wild, C. J.: Nonlinear regression. (1989) · Zbl 0721.62062
[43] Sen, M.; Stoffa, P. L.: Global optimization methods in geophysical inversion. (1995) · Zbl 0871.90107
[44] Shubert, B. O.: A sequential method seeking the global maximum of a function. SIAM J. Numer. anal. 9, 379-388 (1972) · Zbl 0251.65052
[45] Szu, H.; Hartley, R.: Fast simulated annealing. Phys. lett. A 122, 157-162 (1987)
[46] Tarantola, A.: Inverse problem theory. (1987) · Zbl 0875.65001
[47] 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)
[48] Tikhonov, A. N.; Arsenin, V. Y.: Solutions of ill-posed problem. (1977) · Zbl 0354.65028
[49] Wolfe, M. A.: An interval algorithm for constrained global optimization. J. comput. Appl. math. 50, 605-612 (1994) · Zbl 0809.65064
[50] Xu, P. L.: A hybrid global optimization methodthe one-dimensional case. J. comput. Appl. math. 147, 301-314 (2002) · Zbl 1013.65063
[51] P.L. Xu, Numerical solution for bounding feasible point sets, J. Comput. Appl. Math., to be published. · Zbl 1022.65068