An outcome space algorithm for minimizing the product of two convex functions       over a convex set

Nguyen Thi Bach Kim; Nguyen Canh Nam; Le Quang Thuy

doi:10.3934/jimo.2013.9.243

Article Contents

2013, Volume 9, Issue 1: 243-253. Doi: 10.3934/jimo.2013.9.243

This issue Previous Article On the Levenberg-Marquardt methods for convex constrained nonlinear equations Next Article Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities

An outcome space algorithm for minimizing the product of two convex functions over a convex set

1.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, N01 Dai Co Viet, Hai Ba Trung, Hanoi

Received: June 2011

Revised: July 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

This paper presents an outcome-space outer approximation algorithm for solving the problem of minimizing the product of two convex functions over a compact convex set in $\mathbb{R}^n$. The computational experiences are reported. The proposed algorithm is convergent.

Keywords:

Global optimization problem,
efficient point,
outcome set,
minimizing the product of two convex functions.

Mathematics Subject Classification: Primary: 90C29; Secondary: 90C26.

Citation:

Related Papers

Cited by

References

[1]	H. P. Benson and G. M. Boger, Multiplicative programming problems: Analysis and efficient point search heuristic, Journal of Optimization Theory and Applications, 94 (1997), 487-510.doi: 10.1023/A:1022600232285.
[2]	H. P. Benson and G. M. Boger, Outcome-space cutting-plane algorithm for linear multiplicative programming, Journal of Optimization Theory and Applications, 104 (2000), 301-322.doi: 10.1023/A:1004657629105.
[3]	H. P. Benson, An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming, Journal of Global Optimization, 15 (1999), 315-342.doi: 10.1023/A:1008316429329.
[4]	Y. Gao, G. Wu and W. Ma, A new global optimization approach for convex multiplicative programming, Applied Mathematics and Computation, 216 (2010), 1206-1218.doi: 10.1016/j.amc.2010.02.012.
[5]	R. Hosrt, N. V. Thoai and J. Devries, On finding the new vertices and redundant constraints in cutting plane algorithms for global optimization, Operations Research Letters, 7 (1988), 85-90.doi: 10.1016/0167-6377(88)90071-5.
[6]	B. Jaumard, C. Meyer and H. Tuy, Generalized convex multiplicative programming via quasiconcave minimization, Journal of Global Optimization, 10 (1997), 229-256.doi: 10.1023/A:1008203116882.
[7]	N. T. B. Kim, Finite algorithm for minimizing the product of two linear functions over a polyhedron, Journal Industrial and Management Optimization, 3 (2007), 481-487.doi: 10.3934/jimo.2007.3.481.
[8]	N. T. B. Kim, N. T. L. Trang and T. T. H. Yen, Outcome-space outer approximation algorithm for linear multiplicative programming, East West Journal of Mathematics, 9 (2007), 81-98.
[9]	H. Konno and T. Kuno, Linear multiplicative programming, Mathematical Programming, 56 (1992), 51-64.doi: 10.1007/BF01580893.
[10]	H. Konno and T. Kuno, Multiplicative programming problems, Handbook of Global Optimization, (Edited by R. Horst and P. M. Pardalos), Kluwer Academic Publishers, Dordrecht, Netherlands, (1995), 369-405.
[11]	D. T. Luc, "Theory of Vector Optimization," Springer-Verlag, Berlin, Germany, 1989.doi: 10.1007/978-3-642-50280-4.
[12]	T. Matsui, NP-hardness of linear multiplicative programming and related problems, Journal of Global Optimization, 9 (1996), 113-119.doi: 10.1007/BF00121658.
[13]	L. D. Muu and B. T. Tam, Minimizing the sum of a convex function and the product of two affine functions over a convex set, Optimization, 24 (1992), 57-62.doi: 10.1080/02331939208843779.
[14]	H. X. Phu, On efficient sets in $\mathbbR^2$, Vietnam Journal of Mathematics, 33 (2005), 463-468.
[15]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.
[16]	T. V. Thieu, A finite method for globally minimizing concave function over unbounded polyhedral convex sets and its applications, Acta Mathematica Hungarica, 52 (1988), 21-36.doi: 10.1007/BF01952475.
[17]	N. V. Thoai, A global optimization approach for solving the convex multiplicative programming problem, Journal of Global Optimization, 1 (1991), 341-357.doi: 10.1007/BF00130830.
[18]	P. L. Yu, "Multiple-Criteria Decision Making," Plenum Press, New York and London, 1985.doi: 10.1007/978-1-4684-8395-6.