# American Institute of Mathematical Sciences

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 & 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.  doi: 10.1080/02331939208843857.  Google Scholar [2] T. Amahroq and A. Taa, On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.  doi: 10.1080/02331939708844332.  Google Scholar [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.  doi: 10.1090/S0002-9947-98-01983-7.  Google Scholar [4] J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications,, Nonlinear Anal., 38 (1999), 687.  doi: 10.1016/S0362-546X(98)00142-4.  Google Scholar [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.  doi: 10.1007/s10957-009-9609-1.  Google Scholar [6] J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Funct. Anal. Optim., 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar [7] J. Dutta, Necessary optimality conditions and saddle points for approximate optimization in Banach spaces,, Top, 13 (2005), 127.  doi: 10.1007/BF02578991.  Google Scholar [8] J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces,, Math. Meth. Oper. Res., 64 (2006), 521.  doi: 10.1007/s00186-006-0079-z.  Google Scholar [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.  doi: 10.1137/05062648X.  Google Scholar [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 (2003).   Google Scholar [11] S. Helbig, On a new concept for $\varepsilon$-efficiency,, talk at, (1992).   Google Scholar [12] J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79.   Google Scholar [13] J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces,", Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics], 31 (1986).   Google Scholar [14] S. S. Kutateladze, Convex $\varepsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391.   Google Scholar [15] P. Loridan, $\varepsilon$-solutions in vector minimization problems,, J. Optim. Theory Appl., 43 (1984), 265.  doi: 10.1007/BF00936165.  Google Scholar [16] P. Loridan, Necessary conditions for $\varepsilon$-optimality,, Math. Programming Study, 19 (1982), 140.  doi: 10.1007/BFb0120986.  Google Scholar [17] B. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces,, Trans. Amer. Math. Soc., 348 (1996), 1235.   Google Scholar [18] B. S. Mordukhovich, Necessary conditions in nonsmooth minimization via lower and upper subgradients,, Set-Valued Anal., 12 (2004), 163.  doi: 10.1023/B:SVAN.0000023398.73288.82.  Google Scholar [19] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory,", Grundlehren der Mathematischen Wissenschaften, (2006).   Google Scholar [20] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. II: Applications,", Grundlehren der Mathematischen Wissenschaften, (2006).   Google Scholar [21] A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.  doi: 10.1016/0362-546X(86)90126-4.  Google Scholar [22] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.   Google Scholar [23] D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar [24] A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.  doi: 10.1137/S0363012902411532.  Google Scholar [25] X. Y. Zheng and K. F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces,, SIAM J. on Optim., 17 (2006), 1154.  doi: 10.1137/060651860.  Google Scholar [26] Q. J. Zhu, Necessary conditions for constrained optimization problems in smooth Banach spaces and applications,, SIAM J. Optim., 12 (2002), 1032.  doi: 10.1137/S105262340138339.  Google Scholar

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##### References:
 [1] B. El Abdouni and L. Thibault, Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces,, Optimization, 26 (1992), 277.  doi: 10.1080/02331939208843857.  Google Scholar [2] T. Amahroq and A. Taa, On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.  doi: 10.1080/02331939708844332.  Google Scholar [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.  doi: 10.1090/S0002-9947-98-01983-7.  Google Scholar [4] J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications,, Nonlinear Anal., 38 (1999), 687.  doi: 10.1016/S0362-546X(98)00142-4.  Google Scholar [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.  doi: 10.1007/s10957-009-9609-1.  Google Scholar [6] J. Dutta and V. Vetrivel, On approximate minima in vector optimization,, Numer. Funct. Anal. Optim., 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar [7] J. Dutta, Necessary optimality conditions and saddle points for approximate optimization in Banach spaces,, Top, 13 (2005), 127.  doi: 10.1007/BF02578991.  Google Scholar [8] J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces,, Math. Meth. Oper. Res., 64 (2006), 521.  doi: 10.1007/s00186-006-0079-z.  Google Scholar [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.  doi: 10.1137/05062648X.  Google Scholar [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 (2003).   Google Scholar [11] S. Helbig, On a new concept for $\varepsilon$-efficiency,, talk at, (1992).   Google Scholar [12] J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces,, Math. Oper. Res., 4 (1979), 79.   Google Scholar [13] J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces,", Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics], 31 (1986).   Google Scholar [14] S. S. Kutateladze, Convex $\varepsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391.   Google Scholar [15] P. Loridan, $\varepsilon$-solutions in vector minimization problems,, J. Optim. Theory Appl., 43 (1984), 265.  doi: 10.1007/BF00936165.  Google Scholar [16] P. Loridan, Necessary conditions for $\varepsilon$-optimality,, Math. Programming Study, 19 (1982), 140.  doi: 10.1007/BFb0120986.  Google Scholar [17] B. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces,, Trans. Amer. Math. Soc., 348 (1996), 1235.   Google Scholar [18] B. S. Mordukhovich, Necessary conditions in nonsmooth minimization via lower and upper subgradients,, Set-Valued Anal., 12 (2004), 163.  doi: 10.1023/B:SVAN.0000023398.73288.82.  Google Scholar [19] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory,", Grundlehren der Mathematischen Wissenschaften, (2006).   Google Scholar [20] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. II: Applications,", Grundlehren der Mathematischen Wissenschaften, (2006).   Google Scholar [21] A. B. Németh, A nonconvex vector minimization problem,, Nonlinear Anal., 10 (1986), 669.  doi: 10.1016/0362-546X(86)90126-4.  Google Scholar [22] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, in, (1995), 497.   Google Scholar [23] D. J. White, Epsilon efficiency,, J. Optim. Theory Appl., 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar [24] A. Zaffaroni, Degrees of efficiency and degrees of minimality,, SIAM J. Control Optim., 42 (2003), 1071.  doi: 10.1137/S0363012902411532.  Google Scholar [25] X. Y. Zheng and K. F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces,, SIAM J. on Optim., 17 (2006), 1154.  doi: 10.1137/060651860.  Google Scholar [26] Q. J. Zhu, Necessary conditions for constrained optimization problems in smooth Banach spaces and applications,, SIAM J. Optim., 12 (2002), 1032.  doi: 10.1137/S105262340138339.  Google Scholar
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