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New filled functions for nonsmooth global optimization. (English) Zbl 1205.90236
Summary: The filled function method is an effective approach to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. This paper gives a new definition for the filled function, which overcomes some drawbacks of the previous definition. It proposes a two-parameter filled function and a one-parameter filled function to improve the efficiency of numerical computation. Based on these analyses, two corresponding filled function algorithms are presented. They are global optimization methods which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods.

MSC:
90C26 Nonconvex programming, global optimization
90C56 Derivative-free methods and methods using generalized derivatives
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[1] Sadegheih, A., Scheduling problem using genetic algorithm, simulated annealing and the effects of parameter values on GA performance, Appl. math. model., 30, 2, 147-154, (2008) · Zbl 1163.90511
[2] Birbil, S.I.; Fang, S.C., An electromagnetism-like mechanism for global optimization, J. global optim., 25, 3, 263-282, (2003) · Zbl 1047.90045
[3] Chen, F.J., Exponential stability of delayed cellular neural networks, J. zhejiang normal university (natural sciences), 30, 1, 34-38, (2007) · Zbl 1174.34441
[4] Wu, Z.Y.; Zhang, L.S.; Teo, K.L., New modified function for global optimization, J. optim. theory appl., 25, 3, 181-203, (2005) · Zbl 1114.90100
[5] Levy, A.V.; Montalvo, A., The tunneling algorithm for the global minimization of functions, SIAM J. sci. statist. comput., 6, 1, 15-29, (1985) · Zbl 0601.65050
[6] Ratz, D., A nonsmooth global optimization technique using slopes: one-dimensional case, J. global optim., 14, 4, 365-393, (1999) · Zbl 0959.65078
[7] Csendes, T.; Pinter, J., The impact of accelerating tools on the interval subdivision algorithm for global optimization, Eur. J. oper. res., 65, 314-320, (1993) · Zbl 0768.90068
[8] Ge, R.P., The theory of filled function methods for finding global minimizers of nonlinearly constrained minimization problems, J. comput. math., 5, 1-9, (1987)
[9] Ge, R.P.; Qin, Y.F., The globally convexized filled functions for global optimization, Appl. math. comput., 35, 2, 131-158, (1990) · Zbl 0752.65052
[10] Han, Q.M.; Han, J.Y., Revised filled function methods for global optimization, Appl. math. comput., 119, 2, 217-228, (2001) · Zbl 1053.90111
[11] Zhang, L.S.; Ng, C.K.; Li, D., A new filled function method for global optimization, J. global optim., 28, 1, 17-43, (2004) · Zbl 1061.90109
[12] Zhang, L.S.; Li, D., Global search in nonlinear integer programming: filled function approach, Int. confer. optim. technol. appl. Perth, 1998, 446-452, (1999)
[13] Wang, Daqing; Sheng, Songbai, A modified filled function method for finding a global minimizer of a function of several variables, J. comp. math. suppl. issue, 60-65, (1992) · Zbl 0794.65059
[14] Shang, Y.; Pu, D.; Jiang, A., Finding global minimizer with one-parameter filled function on unconstrained global optimization, Appl. math. comput., 191, 1, 176-182, (2007) · Zbl 1193.90174
[15] Shor, N., Minimization methods for non-differentiable function, (1985), Springer-Verlag Berlin · Zbl 0561.90058
[16] Fuduli, A.; Gaudioso, M.; Giallombardo, G., Minimizing nonconvex nonsmooth functions via cutting planes and proximity control, SIAM J. optim., 14, 3, 743-756, (2003) · Zbl 1079.90105
[17] Kelley, J.E., The cutting-plane method for solving convex programs, J. soc. ind. appl. math., 8, 4, 703-712, (1960) · Zbl 0098.12104
[18] Goffin, J.L.; Haurie, A.; Vial, J.P., Decomposition and nondifferentiable optimization with the projective algorithm, Manag. sci., 38, 2, 284-302, (1992) · Zbl 0762.90050
[19] Solodov, M.V., A bundle method for a class of bilevel nonsmooth convex minimization problems, SIAM J. optim., 18, 242-259, (2007) · Zbl 1145.90082
[20] Schramm, H.; Zowe, J.A., A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results, SIAM J. optim., 2, 1, 121-152, (1992) · Zbl 0761.90090
[21] Ardenghi, J.I.; Maciel, M.C.; Verdiell, A.B., A trust-region-approach for solving a parameter estimation problem from the biotechnology area, Appl. numer. math., 47, 3, 281-294, (2003) · Zbl 1030.92022
[22] Zowe, J., The BT-algorithm for minimizing a nonsmooth functional subject to linear constrains, (), 459-480 · Zbl 0735.90069
[23] Zhu, W.X., A class of filled functions irrelevant to the number of local minimizers for global optimization, J. syst. sci. math. sci., 22, 4, 406-413, (2002) · Zbl 1111.90370
[24] Kong, M., On the filled function method for nonsmooth program, J. syst. sci. math. sci., 20, 4, 149-154, (2004)
[25] Wu, Q.; Liu, S.Y.; Zhang, L.Y., A modified filled function method for a global minimizer of a nonsmooth programming problem, Math. appl., 17, 2, 36-40, (2004)
[26] Yao, Y., Dynamic tunneling algorithm for global optimization, IEEE trans syst. man cybern., 19, 5, 1222-1230, (1989)
[27] Shen, Z.H.; Zhu, Y., An interval version of subbert’s iterative method for the localization of the global maximum, Computing, 38, 275-280, (1987) · Zbl 0602.65040
[28] Gorges, C.; Ratschek, H., Global interval methods for local nonsmooth optimization, J. global optim., 14, 2, 157-179, (1999) · Zbl 0946.90085
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