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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

 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

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.
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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [9] H. Konno and T. Kuno, Linear multiplicative programming,, Mathematical Programming, 56 (1992), 51.  doi: 10.1007/BF01580893.  Google Scholar [10] H. Konno and T. Kuno, Multiplicative programming problems,, Handbook of Global Optimization, (1995), 369.   Google Scholar [11] D. T. Luc, "Theory of Vector Optimization,", Springer-Verlag, (1989).  doi: 10.1007/978-3-642-50280-4.  Google Scholar [12] T. Matsui, NP-hardness of linear multiplicative programming and related problems,, Journal of Global Optimization, 9 (1996), 113.  doi: 10.1007/BF00121658.  Google Scholar [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.  Google Scholar [14] H. X. Phu, On efficient sets in $\mathbbR^2$,, Vietnam Journal of Mathematics, 33 (2005), 463.   Google Scholar [15] R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] P. L. Yu, "Multiple-Criteria Decision Making,", Plenum Press, (1985).  doi: 10.1007/978-1-4684-8395-6.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [9] H. Konno and T. Kuno, Linear multiplicative programming,, Mathematical Programming, 56 (1992), 51.  doi: 10.1007/BF01580893.  Google Scholar [10] H. Konno and T. Kuno, Multiplicative programming problems,, Handbook of Global Optimization, (1995), 369.   Google Scholar [11] D. T. Luc, "Theory of Vector Optimization,", Springer-Verlag, (1989).  doi: 10.1007/978-3-642-50280-4.  Google Scholar [12] T. Matsui, NP-hardness of linear multiplicative programming and related problems,, Journal of Global Optimization, 9 (1996), 113.  doi: 10.1007/BF00121658.  Google Scholar [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.  Google Scholar [14] H. X. Phu, On efficient sets in $\mathbbR^2$,, Vietnam Journal of Mathematics, 33 (2005), 463.   Google Scholar [15] R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] P. L. Yu, "Multiple-Criteria Decision Making,", Plenum Press, (1985).  doi: 10.1007/978-1-4684-8395-6.  Google Scholar

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