Abstract
This paper discusses algorithms of Moore, Skelboe, Ichida, Fujii and Hansen for solving the global unconstrained optimization problem. These algorithms have been tried on computers, but a thorough theoretical discussion of their convergence properties has been missing. The discussion was started in part I of this paper (Mathematical Programming 33 (1985) 300–317) where the convergence to the global minimum was studied. The present paper is concerned with the different behaviours of these algorithms when they are used for the determination of global minimum points. The solution sets of the algorithms can be a subset of the set of global minimum points,G, a superset ofG, or exactlyG. The algorithms are applicable to a very general class of functions: functions which are continuous, and have suitable inclusion functions. The number of global minimum points can be infinite.
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References
N.S. Asaithambi, Z. Shen and R.E. Moore, “On computing the range of values,”Computing 28 (1982) 225–237.
P. Basso, “Iterative methods for the localisation of the global maximum,”SIAM Journal of Numerical Analysis 19 (1982) 781–792.
H.P. Benson, “On the convergence of two branch-and-bounds algorithms for nonconvex programming problems,”Journal of Optimization Theory and Applications 36(1982) 129–134.
E. Hansen, “Global optimisation using interval analysis: the one-dimensional case,”Jorunal of Optimization Theory and Applications 29(1979) 331–334.
E. Hansen, “Global optimisation using interval analysis—the multi-dimensional case,”Numerische Mathematik 34 (1980) 247–270.
K. Ichida and Y. Fujii, “An interval arithmetic method for global optimization,” Computing 23(1979) 85–97.
R.E. Moore, Interval analysis (Prentice-Hall, Englewood Cliffs, 1966).
R.E. Moore, “On computing the range of a ration function ofn variables over a bounded region,”Computing 16(1976) 1–15.
R.E. Moore,Methods and Applications of Interval Analysis (Sian, Philadelphia, 1979).
L.B. Rall, “Global oprimization using automatic differentiation and interval iteration,” Technical Summary Report #2832, Mathematics Research Center, University of Wisconsin-Madison (1985).
H. Ratschek, “Inclusion functions and global optimization,”Mathematical Programming 33(1985) 300–317.
H. Ratschek and J. Ronke,Computer Methods for the Range of Functions (Horwood, Chichester, 1984).
S. Skelboe, “Computation of rational interval functions,”Nordisk Tidskrift for Informationsbehandling (BIT) 14 (1974) 85–97.
G.W. Walster, E.R. Hansen and S. Sengupta, “Test results for a global optimization algorithm,” in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization 1984 (SIAM, Philadelphia, 1985) pp. 272–287.
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This work was supported by the Deutsche Forschungsgemeinschaft.
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Moore, R.E., Ratschek, H. Inclusion functions and global optimization II. Mathematical Programming 41, 341–356 (1988). https://doi.org/10.1007/BF01580772
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DOI: https://doi.org/10.1007/BF01580772