A new linearization method for generalized linear multiplicative programming. (English) Zbl 1208.90162

Summary: This paper presents a deterministic global optimization algorithm for solving generalized linear multiplicative programming (GLMP). In this algorithm, a new linearization method is proposed, which applies more information of the function of (GLMP) than some other methods. By using this new linearization technique, the initial nonconvex problem is reduced to a sequence of linear programming problems. A deleting rule is presented to improve the convergence speed of this algorithm. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency of this algorithm.


90C30 Nonlinear programming
90C26 Nonconvex programming, global optimization
Full Text: DOI


[1] Avriel, M.; Diewert, W.E.; Schaible, S.; Zhang, I., Generalized convexity, (1988), Plenum Press New York
[2] Horst, R.; Tuy, H., Global optimization: deterministic approaches, (1993), Springer-Verlag Berlin
[3] Henderson, J.M.; Quandt, R.E., Microeconomic theory, (1971), McGraw-Hill New York · Zbl 0224.90014
[4] Konno, H.; Shirakawa, H.; Yamazaki, H., A Mean-absolute deviation-skewness portfolio optimization model, Annals of operations research, 45, 205-220, (1993) · Zbl 0785.90014
[5] Dorneich, M.C.; Sahinidis, N.V., Global optimization algorithms for chip design and compaction, Engineering optimization, 25, 131-154, (1995)
[6] Bennett, K.P., Global tree optimization: a non-greedy decision tree algorithm, Computing sciences and statistics, 26, 156-160, (1994)
[7] Keeney, R.L.; Raiffa, H., Decisions with multiple objective, (1993), Cambridge University Press Cambridge, MA · Zbl 0488.90001
[8] Mulvey, J.M.; Vanderbei, R.J.; Zenios, S.A., Robust optimization of large-scale systems, Operations research, 43, 264-281, (1995) · Zbl 0832.90084
[9] Benson, H.P., Decomposition branch and bound based algorithm for linear programs with additional multiplicative constraints, Journal of optimization theory and applications, 126, 41-46, (2005) · Zbl 1093.90040
[10] Ryoo, H.S.; Sahinidis, N.V., Global optimization of multiplicative programs, Journal of global optimization, 26, 387-418, (2003) · Zbl 1052.90091
[11] Kuno, T., A finite branch-and-bound algorithm for linear multiplicative programming, Computational optimization and application, 20, 119-135, (2001) · Zbl 0983.90075
[12] Gao, Y.L.; Xu, C.X.; Yang, Y.J., An out come-space finite algorithm for solving linear multiplicative programming, Applied mathematics and computation, 179, 494-505, (2006) · Zbl 1103.65065
[13] Benson, H.P.; Boger, G.M., Outcome-space cutting-plane algorithm for linear multiplicative programming, Journal of optimization theory and applications, 104, 301-322, (2000) · Zbl 0962.90024
[14] Liu, X.J.; Umegaki, T.; Yamamoto, Y., Heuristic methods for linear multiplicative programming, Journal of global optimization, 4, 433-447, (1999) · Zbl 0966.90051
[15] Falk, J.E.; Palocsa, S.W., Image space analysis of generalized fractional programs, Journal of global optimization, 4, 63-88, (1994) · Zbl 0809.90121
[16] Shen, P.P.; Jiao, H.W., Linearization method for a class of multiplicative programming with exponent, Applied mathematics and computation, 183, 328-333, (2006) · Zbl 1110.65051
[17] Jiao, H.W., A branch and bound algorithm for globally solving a class of nonconvex programming problems, Nonlinear analysis: theory, methods and applications, 70, 1113-1123, (2008) · Zbl 1155.90459
[18] Shen, P.P.; Bai, X.D.; Li, W.M., A new accelerating method for globally solving a class of nonconvex programming problems, Nonlinear analysis: theory, methods and applications, 71, 2866-2876, (2009) · Zbl 1168.90576
[19] Zhou, X.G.; Wu, K., A method of acceleration for a class of multiplicative programming problems with exponent, Journal of computational and applied mathematics, 223, 975-982, (2009) · Zbl 1159.65067
[20] Horst, R.; Thoai, N.V., DC programming: overview, Journal of global optimization, 103, 1-43, (1999) · Zbl 1073.90537
[21] Bittner, L., Some representation theorems for function and sets and their application to nonlinear programming, Numerische Mathematik, 16, 32-51, (1970) · Zbl 0217.27602
[22] Blanquero, R.; Carrizosa, E., On covering methods for d.c. optimization, Journal of global optimization, 18, 265-274, (2000) · Zbl 1039.90055
[23] Blanquero, R.; Carrizosa, E.; Conde, E., Finding GM-estimators with global optimization techniques, Journal of global optimization, 21, 223-237, (2000) · Zbl 1021.90042
[24] Tuy, H., D.C. optimization: theory, methods and algorithms, ()
[25] Hoai An, L.T.; Belghiti, M.T.; Tao, P.D., A new efficient algorithm based on DC programming and DCA for clustering, Journal of global optimization, 37, 593-608, (2007) · Zbl 1198.90327
[26] Phuong, N.T.H.; Tuy, H., A unified monotonic approach to generalized linear fractional programming, Journal of global optimization, 26, 229-259, (2003) · Zbl 1039.90079
[27] Tuy, H., Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms, Journal of global optimization, 1, 23-36, (1991) · Zbl 0744.90083
[28] Thoai, N.V., A global optimization approach for solving the convex multiplicative programming problems, Journal of global optimization, 1, 341-357, (1991) · Zbl 0752.90056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.