July  2012, 8(3): 749-764. doi: 10.3934/jimo.2012.8.749

Lagrange multiplier rules for approximate solutions in vector optimization

1. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Received  January 2012 Revised  March 2012 Published  June 2012

In Asplund space, Lagrange multiplier rules for approximate solutions of nonsmooth vector optimization problems are studied. The relationships between the vector and the scalar optimization problems are established. And the optimality conditions of approximate solutions for vector optimization are obtained. Moreover, the vector variational inequalities are considered by applying the partial results given in this paper.
Citation: Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial and Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
References:
[1]

B. El Abdouni and L. Thibault, Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces, Optimization, 26 (1992), 277-285. doi: 10.1080/02331939208843857.

[2]

T. Amahroq and A. Taa, On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems, Optimization, 41 (1997), 159-172. doi: 10.1080/02331939708844332.

[3]

J. M. Borwein, J. S. Treiman and Q. J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc., 350 (1998), 2409-2429. doi: 10.1090/S0002-9947-98-01983-7.

[4]

J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Anal., 38 (1999), 687-773. doi: 10.1016/S0362-546X(98)00142-4.

[5]

M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\varepsilon$-Pareto solutions in vector optimization with non-solid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211. doi: 10.1007/s10957-009-9609-1.

[6]

J. Dutta and V. Vetrivel, On approximate minima in vector optimization, Numer. Funct. Anal. Optim., 22 (2001), 845-859. doi: 10.1081/NFA-100108312.

[7]

J. Dutta, Necessary optimality conditions and saddle points for approximate optimization in Banach spaces, Top, 13 (2005), 127-143. doi: 10.1007/BF02578991.

[8]

J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces, Math. Meth. Oper. Res., 64 (2006), 521-540. doi: 10.1007/s00186-006-0079-z.

[9]

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), 688-710. doi: 10.1137/05062648X.

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, 17, Springer-Verlag, New York, 2003.

[11]

S. Helbig, On a new concept for $\varepsilon$-efficiency, talk at "Optimization Days, 1992," Montreal, 1992.

[12]

J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.

[13]

J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics], 31, Verlag Peter D. Lang, Frankfurt am Main, 1986.

[14]

S. S. Kutateladze, Convex $\varepsilon$-programming, Soviet Math. Dokl., 20 (1979), 391-393.

[15]

P. Loridan, $\varepsilon$-solutions in vector minimization problems, J. Optim. Theory Appl., 43 (1984), 265-276. doi: 10.1007/BF00936165.

[16]

P. Loridan, Necessary conditions for $\varepsilon$-optimality, Math. Programming Study, 19 (1982), 140-152. doi: 10.1007/BFb0120986.

[17]

B. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348 (1996), 1235-1280.

[18]

B. S. Mordukhovich, Necessary conditions in nonsmooth minimization via lower and upper subgradients, Set-Valued Anal., 12 (2004), 163-193. doi: 10.1023/B:SVAN.0000023398.73288.82.

[19]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory," Grundlehren der Mathematischen Wissenschaften, Vol. 330, Springer-Verlag, Berlin, 2006.

[20]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. II: Applications," Grundlehren der Mathematischen Wissenschaften, Vol. 331, Springer-Verlag, Berlin, 2006.

[21]

A. B. Németh, A nonconvex vector minimization problem, Nonlinear Anal., 10 (1986), 669-678. doi: 10.1016/0362-546X(86)90126-4.

[22]

T. Tanaka, A new approach to approximation of solutions in vector optimization problems, in "Proceedings of APORS, 1994" (eds. M. Fushimi and K. Tone), World Scientific Publishing, Singapore, (1995), 497-504.

[23]

D. J. White, Epsilon efficiency, J. Optim. Theory Appl., 49 (1986), 319-337. doi: 10.1007/BF00940762.

[24]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086. doi: 10.1137/S0363012902411532.

[25]

X. Y. Zheng and K. F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. on Optim., 17 (2006), 1154-1175. doi: 10.1137/060651860.

[26]

Q. J. Zhu, Necessary conditions for constrained optimization problems in smooth Banach spaces and applications, SIAM J. Optim., 12 (2002), 1032-1047. doi: 10.1137/S105262340138339.

show all references

References:
[1]

B. El Abdouni and L. Thibault, Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces, Optimization, 26 (1992), 277-285. doi: 10.1080/02331939208843857.

[2]

T. Amahroq and A. Taa, On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems, Optimization, 41 (1997), 159-172. doi: 10.1080/02331939708844332.

[3]

J. M. Borwein, J. S. Treiman and Q. J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc., 350 (1998), 2409-2429. doi: 10.1090/S0002-9947-98-01983-7.

[4]

J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Anal., 38 (1999), 687-773. doi: 10.1016/S0362-546X(98)00142-4.

[5]

M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\varepsilon$-Pareto solutions in vector optimization with non-solid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211. doi: 10.1007/s10957-009-9609-1.

[6]

J. Dutta and V. Vetrivel, On approximate minima in vector optimization, Numer. Funct. Anal. Optim., 22 (2001), 845-859. doi: 10.1081/NFA-100108312.

[7]

J. Dutta, Necessary optimality conditions and saddle points for approximate optimization in Banach spaces, Top, 13 (2005), 127-143. doi: 10.1007/BF02578991.

[8]

J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces, Math. Meth. Oper. Res., 64 (2006), 521-540. doi: 10.1007/s00186-006-0079-z.

[9]

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), 688-710. doi: 10.1137/05062648X.

[10]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces," CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, 17, Springer-Verlag, New York, 2003.

[11]

S. Helbig, On a new concept for $\varepsilon$-efficiency, talk at "Optimization Days, 1992," Montreal, 1992.

[12]

J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.

[13]

J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics], 31, Verlag Peter D. Lang, Frankfurt am Main, 1986.

[14]

S. S. Kutateladze, Convex $\varepsilon$-programming, Soviet Math. Dokl., 20 (1979), 391-393.

[15]

P. Loridan, $\varepsilon$-solutions in vector minimization problems, J. Optim. Theory Appl., 43 (1984), 265-276. doi: 10.1007/BF00936165.

[16]

P. Loridan, Necessary conditions for $\varepsilon$-optimality, Math. Programming Study, 19 (1982), 140-152. doi: 10.1007/BFb0120986.

[17]

B. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348 (1996), 1235-1280.

[18]

B. S. Mordukhovich, Necessary conditions in nonsmooth minimization via lower and upper subgradients, Set-Valued Anal., 12 (2004), 163-193. doi: 10.1023/B:SVAN.0000023398.73288.82.

[19]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory," Grundlehren der Mathematischen Wissenschaften, Vol. 330, Springer-Verlag, Berlin, 2006.

[20]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. II: Applications," Grundlehren der Mathematischen Wissenschaften, Vol. 331, Springer-Verlag, Berlin, 2006.

[21]

A. B. Németh, A nonconvex vector minimization problem, Nonlinear Anal., 10 (1986), 669-678. doi: 10.1016/0362-546X(86)90126-4.

[22]

T. Tanaka, A new approach to approximation of solutions in vector optimization problems, in "Proceedings of APORS, 1994" (eds. M. Fushimi and K. Tone), World Scientific Publishing, Singapore, (1995), 497-504.

[23]

D. J. White, Epsilon efficiency, J. Optim. Theory Appl., 49 (1986), 319-337. doi: 10.1007/BF00940762.

[24]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086. doi: 10.1137/S0363012902411532.

[25]

X. Y. Zheng and K. F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. on Optim., 17 (2006), 1154-1175. doi: 10.1137/060651860.

[26]

Q. J. Zhu, Necessary conditions for constrained optimization problems in smooth Banach spaces and applications, SIAM J. Optim., 12 (2002), 1032-1047. doi: 10.1137/S105262340138339.

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