A filled function method for global optimization. (English) Zbl 1124.65052

Liang, Y. M.; Zhang, L. S.; Li, M. M.; Han, B. S.

J. Comput. Appl. Math. 205, No. 1, 16-31 (2007).

A new filled function method with one parameter for finding a global minimizer for a general class of nonlinear programming problems with a closed bounded box is presented. The concept of the filled functions was introduced in the paper by R. P. Ge [Math. Program., Ser. 46, 191–204 (1990; Zbl 0694.90083)]. Adopting the concept of filled functions, a global optimization problem can be solved via a two-phase cycle. A new algorithm is presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.

Reviewer: Nada Djuranović-Miličić (Belgrade)

MSC:

65K05	Numerical mathematical programming methods
90C11	Mixed integer programming
90C25	Convex programming
90C30	Nonlinear programming

Keywords:

local minimizer; global optimization; filled function method; nonlinear programming; numerical results

References:

[1]	Barhen, J.; Protopopescu, V.; Reister, D., TRUST: a deterministic algorithm for global optimization, Science, 276, 1094-1097, (1977) · Zbl 1226.90073
[2]	Cetin, B.C.; Barhen, J.; Burdick, J.W., Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, J. optim. appl., 77, 97-126, (1993) · Zbl 0801.49001
[3]	Cvijovic‘, D.; Klinowski, J., Taboo search: an approach to the multiple minima problem, Science, 267, 664-666, (1995) · Zbl 1226.90101
[4]	Dixon, L.C.W.; Gomulka, J.; Herson, S.E., Reflection on global optimization problems, (), 398-435
[5]	Ge, R.P., A filled function method for finding a global minimizer of a function of several variables, Math. programming, 46, 191-204, (1990) · Zbl 0694.90083
[6]	Ge, R.P.; Qin, Y.F., A class of filled functions for finding global minimizers of a function of several variables, J. optim. theory appl., 54, 241-252, (1987) · Zbl 0595.65072
[7]	Horst, R.; Pardalos, P.M.; Thoai, N.V., Introduction to global optimization, (1995), Kluwer Academic Publishers Dordrecht · Zbl 0836.90134
[8]	Horst, R.; Tuy, H., Global optimization: deterministic approaches, (1993), Springer Heidelberg
[9]	Levy, A.V.; Montalvo, A., The tunneling algorithm for the global minimization of functions, SIAM J. sci. statist. comput., 6, 15-29, (1985) · Zbl 0601.65050
[10]	Litinetski, V.V.; Abramzon, B.M., MARS—a multi-start adaptive random search method for global constrained optimization in engineering applications, Eng. optim., 30, 125-154, (1998)
[11]	Liu, X., Finding global minima with a computable filled function, J. global optim., 19, 151-161, (2001) · Zbl 1033.90088
[12]	Pardalos, P.M.; Rosen, J.B., Constrained global optimization: algorithms and applications, (1987), Springer Berlin · Zbl 0638.90064
[13]	Rinnoy Kan, A.H.G.; Timmer, G.T., Global optimization, (), 631-662 · Zbl 0525.90076
[14]	Xu, Z.; Huang, H.X.; Pardalos, P.M.; Xu, C.X., Filled functions for unconstrained global optimization, J. global optim., 20, 49-65, (2001) · Zbl 1049.90092
[15]	Yao, Y., Dynamic tunneling algorithm for global optimization, IEEE trans. systems man and cybernet., 19, 1222-1230, (1989)
[16]	Zhang, L.S.; NG, C.K.; Li, D.; Tian, W.W., A new filled function method for global optimization, J. global optim., 28, 17-43, (2004) · Zbl 1061.90109
[17]	Zheng, Q.; Zhuang, D., Integral global minimization: algorithms, implementations and numerical tests, J. global optim., 7, 421-454, (1995) · Zbl 0846.90105