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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

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

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.
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity,, J. Convex Anal., 8 (2001), 71.   Google Scholar

[5]

J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57.  doi: 10.1137/0315004.  Google Scholar

[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).   Google Scholar

[7]

J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).   Google Scholar

[11]

D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532.  doi: 10.1080/01630560802099274.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992).   Google Scholar

[20]

M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387.  doi: 10.1007/BF00934353.  Google Scholar

[21]

J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.   Google Scholar

[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.  Google Scholar

[23]

S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391.   Google Scholar

[24]

H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.   Google Scholar

[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.  Google Scholar

[26]

Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161.  doi: 10.1080/02331939808844406.  Google Scholar

[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.  Google Scholar

[28]

A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.  doi: 10.1016/0362-546X(86)90126-4.  Google Scholar

[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.  Google Scholar

[30]

W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness,, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609.   Google Scholar

[31]

T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.   Google Scholar

[32]

I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435.  doi: 10.1007/BF00941179.  Google Scholar

[33]

D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar

[34]

A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.  doi: 10.1137/S0363012902411532.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

S. Bolintinéanu, Vector variational principles: $\epsilon-$efficiency and scalar stationarity,, J. Convex Anal., 8 (2001), 71.   Google Scholar

[5]

J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM J. Control Optim., 15 (1977), 57.  doi: 10.1137/0315004.  Google Scholar

[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).   Google Scholar

[7]

J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Func. Anal. Optim., 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", Springer-Verlag, (2003).   Google Scholar

[11]

D. Gupta and A. Mehra, Two types of approximate saddle points,, Numer. Func. Anal. Optim., 29 (2008), 532.  doi: 10.1080/01630560802099274.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Helbig, "On a new concept for $\epsilon$-efficency,", A Talk at Optimization Days 1992, (1992).   Google Scholar

[20]

M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387.  doi: 10.1007/BF00934353.  Google Scholar

[21]

J. B. Hiriart-Urruty, New concepts in nondifferentiable programming,, Bull. Soc. Math. France M\'em., 60 (1979), 57.   Google Scholar

[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.  Google Scholar

[23]

S. Kutateladze, Convex $\epsilon-$programming,, Soviet Math. Dokl., 20 (1979), 391.   Google Scholar

[24]

H. W. Kuhn and A. W. Tucker, Nonlinear programming,, from, (1951), 481.   Google Scholar

[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.  Google Scholar

[26]

Z. Li and S. Wang, $\epsilon$-approximate solutions in multiobjective optimization,, Optimization, 44 (1998), 161.  doi: 10.1080/02331939808844406.  Google Scholar

[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.  Google Scholar

[28]

A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.  doi: 10.1016/0362-546X(86)90126-4.  Google Scholar

[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.  Google Scholar

[30]

W. D. Rong, $\epsilon-$efficiency in vector optimization problems with cone subconvexlikeness,, Acta Sci. Natur. Univ. NeiMongol., 28 (1997), 609.   Google Scholar

[31]

T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.   Google Scholar

[32]

I. Vályi, Approximate saddle-point theorems in vector optimization,, J. Optim. Theory Appl., 55 (1987), 435.  doi: 10.1007/BF00941179.  Google Scholar

[33]

D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar

[34]

A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.  doi: 10.1137/S0363012902411532.  Google Scholar

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