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

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January 2013, 9(1): 243-253. doi: 10.3934/jimo.2013.9.243

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

Nguyen Thi Bach Kim ^1, , Nguyen Canh Nam ^1, and Le Quang Thuy ^1,

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

Received June 2011 Revised July 2012 Published December 2012

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

Mathematics Subject Classification: Primary: 90C29; Secondary: 90C2.

Citation: Nguyen Thi Bach Kim, Nguyen Canh Nam, Le Quang Thuy. An outcome space algorithm for minimizing the product of two convex functions over a convex set. Journal of Industrial & Management Optimization, 2013, 9 (1) : 243-253. doi: 10.3934/jimo.2013.9.243

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. 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. 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. 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. 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. 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. 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. 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.
[9]	H. Konno and T. Kuno, Linear multiplicative programming,, Mathematical Programming, 56 (1992), 51. doi: 10.1007/BF01580893.
[10]	H. Konno and T. Kuno, Multiplicative programming problems,, Handbook of Global Optimization, (1995), 369.
[11]	D. T. Luc, "Theory of Vector Optimization,", Springer-Verlag, (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. 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. doi: 10.1080/02331939208843779.
[14]	H. X. Phu, On efficient sets in $\mathbbR^2$,, Vietnam Journal of Mathematics, 33 (2005), 463.
[15]	R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (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. 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. doi: 10.1007/BF00130830.
[18]	P. L. Yu, "Multiple-Criteria Decision Making,", Plenum Press, (1985). doi: 10.1007/978-1-4684-8395-6.

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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. 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. 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. 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. 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. 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. 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. 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.
[9]	H. Konno and T. Kuno, Linear multiplicative programming,, Mathematical Programming, 56 (1992), 51. doi: 10.1007/BF01580893.
[10]	H. Konno and T. Kuno, Multiplicative programming problems,, Handbook of Global Optimization, (1995), 369.
[11]	D. T. Luc, "Theory of Vector Optimization,", Springer-Verlag, (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. 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. doi: 10.1080/02331939208843779.
[14]	H. X. Phu, On efficient sets in $\mathbbR^2$,, Vietnam Journal of Mathematics, 33 (2005), 463.
[15]	R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (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. 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. doi: 10.1007/BF00130830.
[18]	P. L. Yu, "Multiple-Criteria Decision Making,", Plenum Press, (1985). doi: 10.1007/978-1-4684-8395-6.

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