Optimality conditions for approximate solutions of vector optimization problems

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April 2011, 7(2): 483-496. doi: 10.3934/jimo.2011.7.483

Optimality conditions for approximate solutions of vector optimization problems

Ying Gao ^1, , Xinmin Yang ^1, and Kok Lay Teo ^2,

1.	Department of Mathematics, Chongqing Normal University, Chongqing 400047, China
2.	Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845

Received October 2009 Revised March 2011 Published April 2011

Abstract
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In this paper, we introduce a new kind of properly approximate efficient solution of vector optimization problems. Some properties for this new class of approximate solutions are established. Also necessary and sufficient conditions via nonlinear scalarizations are obtained for properly approximate solutions. And under the assumption of cone subconvexlike functions, we derive linear scalarizations for properly approximate efficient solutions.

Keywords: nonlinear scalar functions, scalarization., approximate solutions, cone subconvexlike, Vector optimization problem.

Mathematics Subject Classification: 90C29, 90C30, 90C4.

Citation: Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

References:

[1]	E. M. Bednarczuk and M. J. Przybyla, The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors,, SIAM J. Optim., 18 (2007), 907. doi: 10.1137/060658989.
[2]	M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization,, Bull. Math. Soc. Sci. Math. Roumanie Tome, 99 (2008), 109.
[3]	H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9.
[4]	S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity,, J. Convex Anal., 8 (2001), 71.
[5]	J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57. doi: 10.1137/0315004.
[6]	G. Y. Chen, X. X. Huang and X. M. Yang, "Vector Optimization. Set-Valued and Variational Analysis,", Lecture Notes in Econom. and Math. Systems \textbf{541}, 541 (2005).
[7]	J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845. doi: 10.1081/NFA-100108312.
[8]	M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces,, J. Optim. Theory Appl., 145 (2010), 196. doi: 10.1007/s10957-009-9609-1.
[9]	J. B. G. Frenk and G. Kassay, On classes of generalized convex functions, Gordan-Farkas type theorems, and Lagrangian duality,, J. Optim. Theory Appl., 102 (1999), 315. doi: 10.1023/A:1021780423989.
[10]	A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).
[11]	D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532. doi: 10.1080/01630560802099274.
[12]	C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM J. Control Optim., 47 (2008), 883. doi: 10.1137/060672868.
[13]	C. Gutiérrez, R. López and V. Novo, Generalized $\epsilon-$quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Anal., 72 (2010), 4331. doi: 10.1016/j.na.2010.02.012.
[14]	C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM J. Optim., 17 (2006), 688. doi: 10.1137/05062648X.
[15]	C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Math. Methods Oper. Res., 64 (2006), 165. doi: 10.1007/s00186-006-0078-0.
[16]	C. Gutiérrez, B. Jiménez and V. Novo, Optimality conditions via scalarization for a new $\epsilon$-efficiency concept in vector optimization problems,, European J. Oper. Res., 201 (2010), 11. doi: 10.1016/j.ejor.2009.02.007.
[17]	A. M. Geoffrion, Proper efficiency and the theory of vector maximization,, J. Math. Anal. Appl., 22 (1968), 618. doi: 10.1016/0022-247X(68)90201-1.
[18]	T. X. D. Ha, The Ekeland variational principle for Henig proper minimizers and super minimizers,, J. Math. Anal. Appl., 364 (2010), 156. doi: 10.1016/j.jmaa.2009.10.065.
[19]	S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992).
[20]	M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353.
[21]	J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.
[22]	J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79. doi: 10.1287/moor.4.1.79.
[23]	S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391.
[24]	H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.
[25]	J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109. doi: 10.1016/S0893-9659(99)00087-7.
[26]	Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161. doi: 10.1080/02331939808844406.
[27]	C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization,, Math. Program., 119 (2009), 215. doi: 10.1016/j.colsurfa.2009.04.036.
[28]	A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669. doi: 10.1016/0362-546X(86)90126-4.
[29]	W. D. Rong and Y. N. Wu, $\epsilon$-weak minimal solutions of vector optimization problems with set-valued maps,, J. Optim. Theory Appl., 106 (2000), 569. doi: 10.1023/A:1004657412928.
[30]	W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness,, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609.
[31]	T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.
[32]	I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435. doi: 10.1007/BF00941179.
[33]	D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319. doi: 10.1007/BF00940762.
[34]	A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071. doi: 10.1137/S0363012902411532.

show all references

References:

[1]	E. M. Bednarczuk and M. J. Przybyla, The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors,, SIAM J. Optim., 18 (2007), 907. doi: 10.1137/060658989.
[2]	M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization,, Bull. Math. Soc. Sci. Math. Roumanie Tome, 99 (2008), 109.
[3]	H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9.
[4]	S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity,, J. Convex Anal., 8 (2001), 71.
[5]	J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57. doi: 10.1137/0315004.
[6]	G. Y. Chen, X. X. Huang and X. M. Yang, "Vector Optimization. Set-Valued and Variational Analysis,", Lecture Notes in Econom. and Math. Systems \textbf{541}, 541 (2005).
[7]	J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845. doi: 10.1081/NFA-100108312.
[8]	M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces,, J. Optim. Theory Appl., 145 (2010), 196. doi: 10.1007/s10957-009-9609-1.
[9]	J. B. G. Frenk and G. Kassay, On classes of generalized convex functions, Gordan-Farkas type theorems, and Lagrangian duality,, J. Optim. Theory Appl., 102 (1999), 315. doi: 10.1023/A:1021780423989.
[10]	A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).
[11]	D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532. doi: 10.1080/01630560802099274.
[12]	C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM J. Control Optim., 47 (2008), 883. doi: 10.1137/060672868.
[13]	C. Gutiérrez, R. López and V. Novo, Generalized $\epsilon-$quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Anal., 72 (2010), 4331. doi: 10.1016/j.na.2010.02.012.
[14]	C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM J. Optim., 17 (2006), 688. doi: 10.1137/05062648X.
[15]	C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Math. Methods Oper. Res., 64 (2006), 165. doi: 10.1007/s00186-006-0078-0.
[16]	C. Gutiérrez, B. Jiménez and V. Novo, Optimality conditions via scalarization for a new $\epsilon$-efficiency concept in vector optimization problems,, European J. Oper. Res., 201 (2010), 11. doi: 10.1016/j.ejor.2009.02.007.
[17]	A. M. Geoffrion, Proper efficiency and the theory of vector maximization,, J. Math. Anal. Appl., 22 (1968), 618. doi: 10.1016/0022-247X(68)90201-1.
[18]	T. X. D. Ha, The Ekeland variational principle for Henig proper minimizers and super minimizers,, J. Math. Anal. Appl., 364 (2010), 156. doi: 10.1016/j.jmaa.2009.10.065.
[19]	S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992).
[20]	M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353.
[21]	J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.
[22]	J. B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79. doi: 10.1287/moor.4.1.79.
[23]	S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391.
[24]	H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.
[25]	J. C. Liu, $\epsilon-$properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109. doi: 10.1016/S0893-9659(99)00087-7.
[26]	Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161. doi: 10.1080/02331939808844406.
[27]	C. G. Liu, K. F. Ng and W. H. Yang, Merit functions in vector optimization,, Math. Program., 119 (2009), 215. doi: 10.1016/j.colsurfa.2009.04.036.
[28]	A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669. doi: 10.1016/0362-546X(86)90126-4.
[29]	W. D. Rong and Y. N. Wu, $\epsilon$-weak minimal solutions of vector optimization problems with set-valued maps,, J. Optim. Theory Appl., 106 (2000), 569. doi: 10.1023/A:1004657412928.
[30]	W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness,, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609.
[31]	T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.
[32]	I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435. doi: 10.1007/BF00941179.
[33]	D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319. doi: 10.1007/BF00940762.
[34]	A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071. doi: 10.1137/S0363012902411532.

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