# American Institute of Mathematical Sciences

April  2015, 11(2): 673-683. doi: 10.3934/jimo.2015.11.673

## Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps

 1 Department of Mathematics, Chongqing Normal University, Chongqing 400047 2 Department of Mathematics, Chongqing Normal University, Chongqing, 400047 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 4 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hong Kong, China

Received  August 2013 Revised  June 2014 Published  September 2014

In this paper, we characterize approximate solutions of vector optimization problems with set-valued maps. We gives several characterizations of generalized subconvexlike set-valued functions(see [10]), which is a generalization of nearly subconvexlike functions introduced in [34]. We present alternative theorem and derived scalarization theorems for approximate solutions with generalized subconvexlike set-valued maps. And then, Lagrange multiplier theorems under generalized Slater constraint qualification are established.
Citation: Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
##### References:
 [1] T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.  doi: 10.1080/02331939708844332.  Google Scholar [2] S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71.   Google Scholar [3] J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM Journal of Control and Optimization, 15 (1977), 57.  doi: 10.1137/0315004.  Google Scholar [4] G. Y. Chen, X. X. Huang and S. H. Hou, General Ekeland's variational principle for set-valued mappings,, Journal of Optimization Theory and Applications, 106 (2000), 151.  doi: 10.1023/A:1004663208905.  Google Scholar [5] G. Y. Chen and W. D. Rong, Characterization of the benson proper efficiency for nonconvex vector optimization,, Journal of Optimization Theory and Applications, 98 (1998), 365.  doi: 10.1023/A:1022689517921.  Google Scholar [6] M. Ciligot-Travain, On Lagrange Kuhn-Tucker multipliers for Pareto optimization problems,, Numerical Functional Analysis and Optimization, 15 (1994), 689.  doi: 10.1080/01630569408816587.  Google Scholar [7] M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-pareto solutions in vector optimization with nonsolid cones in Banach spaces,, Journal of Optimization Theory and Applications, 145 (2010), 196.  doi: 10.1007/s10957-009-9609-1.  Google Scholar [8] J. Dutta and V. Veterivel, On approximate minima in vector optimization,, Numerical Functional Analysis and Optimization, 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar [9] S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (1992).   Google Scholar [10] Y. Gao, S. H. Hou and X. M. Yang, Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems,, Jornal of Optimization Theory and application, 152 (2012), 97.  doi: 10.1007/s10957-011-9891-6.  Google Scholar [11] Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems,, Journal of Industrial and Management Optimization, 7 (2011), 483.  doi: 10.3934/jimo.2011.7.483.  Google Scholar [12] D. Gupta and A. Mehra, Two types of approximate saddle points,, Numerical Functional Analysis and Optimization, 29 (2008), 532.  doi: 10.1080/01630560802099274.  Google Scholar [13] C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, Journal of Mathematical Analysis and Applications, 389 (2012), 1046.  doi: 10.1016/j.jmaa.2011.12.050.  Google Scholar [14] C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM Journal on Optimization, 17 (2006), 688.   Google Scholar [15] C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165.   Google Scholar [16] C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM Journal on Control And Optimization, 47 (2008), 883.   Google Scholar [17] C. Gutiérrez, B. Jiménez and V. Novo, Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems,, Journal of Global Optimization, 32 (2005), 367.  doi: 10.1007/s10898-004-5904-4.  Google Scholar [18] C. Gutiérrez, R. Lopez and V. Novo, Generalized $\epsilon$-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Analysis, 72 (2010), 4331.  doi: 10.1016/j.na.2010.02.012.  Google Scholar [19] A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137.  doi: 10.1080/02331930108844524.  Google Scholar [20] X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309.  doi: 10.1080/02331930290019440.  Google Scholar [21] Y. W. Huang, Optimality conditions for vector optimization with set-valued maps,, Bulletin of the Australian Mathematical Society 66 (2002), 66 (2002), 317.  doi: 10.1017/S0004972700040168.  Google Scholar [22] V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643.  doi: 10.1080/02331938508843061.  Google Scholar [23] S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048.   Google Scholar [24] Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, Journal of Optimization Theory and Application, 100 (1999), 365.  doi: 10.1023/A:1021786303883.  Google Scholar [25] Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, Journal of Optimization Theory and Applications, 98 (1998), 623.  doi: 10.1023/A:1022676013609.  Google Scholar [26] Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle points in multiobjective programming,, Journal of Optimization Theory and Applications, 83 (1994), 63.  doi: 10.1007/BF02191762.  Google Scholar [27] J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144.  doi: 10.1137/060676465.  Google Scholar [28] J. H. Qiu, A generalized Ekeland vector variational principle and its applications in optimization,, Nonlinear Analysis, 71 (2009), 4705.  doi: 10.1016/j.na.2009.03.034.  Google Scholar [29] J. H. Qiu, Ekeland's variational principle in locally convex spaces and the density of extremal points,, Journal of Mathematical Analysis and Applications, 360 (2009), 317.  doi: 10.1016/j.jmaa.2009.06.054.  Google Scholar [30] P. H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems,, Journal of Optimization Theory and Applications, 119 (2003), 335.  doi: 10.1023/B:JOTA.0000005449.20614.41.  Google Scholar [31] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497.   Google Scholar [32] D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar [33] X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125.   Google Scholar [34] X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, Journal of Optimization Theory and Applications, 110 (2001), 413.  doi: 10.1023/A:1017535631418.  Google Scholar [35] X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627.  doi: 10.1023/A:1026407517917.  Google Scholar

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##### References:
 [1] T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159.  doi: 10.1080/02331939708844332.  Google Scholar [2] S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71.   Google Scholar [3] J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM Journal of Control and Optimization, 15 (1977), 57.  doi: 10.1137/0315004.  Google Scholar [4] G. Y. Chen, X. X. Huang and S. H. Hou, General Ekeland's variational principle for set-valued mappings,, Journal of Optimization Theory and Applications, 106 (2000), 151.  doi: 10.1023/A:1004663208905.  Google Scholar [5] G. Y. Chen and W. D. Rong, Characterization of the benson proper efficiency for nonconvex vector optimization,, Journal of Optimization Theory and Applications, 98 (1998), 365.  doi: 10.1023/A:1022689517921.  Google Scholar [6] M. Ciligot-Travain, On Lagrange Kuhn-Tucker multipliers for Pareto optimization problems,, Numerical Functional Analysis and Optimization, 15 (1994), 689.  doi: 10.1080/01630569408816587.  Google Scholar [7] M. Durea, J. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-pareto solutions in vector optimization with nonsolid cones in Banach spaces,, Journal of Optimization Theory and Applications, 145 (2010), 196.  doi: 10.1007/s10957-009-9609-1.  Google Scholar [8] J. Dutta and V. Veterivel, On approximate minima in vector optimization,, Numerical Functional Analysis and Optimization, 22 (2001), 845.  doi: 10.1081/NFA-100108312.  Google Scholar [9] S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (1992).   Google Scholar [10] Y. Gao, S. H. Hou and X. M. Yang, Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems,, Jornal of Optimization Theory and application, 152 (2012), 97.  doi: 10.1007/s10957-011-9891-6.  Google Scholar [11] Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems,, Journal of Industrial and Management Optimization, 7 (2011), 483.  doi: 10.3934/jimo.2011.7.483.  Google Scholar [12] D. Gupta and A. Mehra, Two types of approximate saddle points,, Numerical Functional Analysis and Optimization, 29 (2008), 532.  doi: 10.1080/01630560802099274.  Google Scholar [13] C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, Journal of Mathematical Analysis and Applications, 389 (2012), 1046.  doi: 10.1016/j.jmaa.2011.12.050.  Google Scholar [14] C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM Journal on Optimization, 17 (2006), 688.   Google Scholar [15] C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165.   Google Scholar [16] C. Gutiérrez, B. Jiménez and V. Novo, A set-valued Ekeland's variational principle in vector optimization,, SIAM Journal on Control And Optimization, 47 (2008), 883.   Google Scholar [17] C. Gutiérrez, B. Jiménez and V. Novo, Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems,, Journal of Global Optimization, 32 (2005), 367.  doi: 10.1007/s10898-004-5904-4.  Google Scholar [18] C. Gutiérrez, R. Lopez and V. Novo, Generalized $\epsilon$-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions,, Nonlinear Analysis, 72 (2010), 4331.  doi: 10.1016/j.na.2010.02.012.  Google Scholar [19] A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137.  doi: 10.1080/02331930108844524.  Google Scholar [20] X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309.  doi: 10.1080/02331930290019440.  Google Scholar [21] Y. W. Huang, Optimality conditions for vector optimization with set-valued maps,, Bulletin of the Australian Mathematical Society 66 (2002), 66 (2002), 317.  doi: 10.1017/S0004972700040168.  Google Scholar [22] V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643.  doi: 10.1080/02331938508843061.  Google Scholar [23] S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048.   Google Scholar [24] Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, Journal of Optimization Theory and Application, 100 (1999), 365.  doi: 10.1023/A:1021786303883.  Google Scholar [25] Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, Journal of Optimization Theory and Applications, 98 (1998), 623.  doi: 10.1023/A:1022676013609.  Google Scholar [26] Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle points in multiobjective programming,, Journal of Optimization Theory and Applications, 83 (1994), 63.  doi: 10.1007/BF02191762.  Google Scholar [27] J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144.  doi: 10.1137/060676465.  Google Scholar [28] J. H. Qiu, A generalized Ekeland vector variational principle and its applications in optimization,, Nonlinear Analysis, 71 (2009), 4705.  doi: 10.1016/j.na.2009.03.034.  Google Scholar [29] J. H. Qiu, Ekeland's variational principle in locally convex spaces and the density of extremal points,, Journal of Mathematical Analysis and Applications, 360 (2009), 317.  doi: 10.1016/j.jmaa.2009.06.054.  Google Scholar [30] P. H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems,, Journal of Optimization Theory and Applications, 119 (2003), 335.  doi: 10.1023/B:JOTA.0000005449.20614.41.  Google Scholar [31] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497.   Google Scholar [32] D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319.  doi: 10.1007/BF00940762.  Google Scholar [33] X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125.   Google Scholar [34] X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, Journal of Optimization Theory and Applications, 110 (2001), 413.  doi: 10.1023/A:1017535631418.  Google Scholar [35] X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627.  doi: 10.1023/A:1026407517917.  Google Scholar
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