Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: 90C29, 90C30, 90C46.

 Citation:

•  [1] T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems, Optimization, 41 (1997), 159-172.doi: 10.1080/02331939708844332. [2] S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity, Journal of Convex Analysis, 8 (2001), 71-85. [3] J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM Journal of Control and Optimization, 15 (1977), 57-63.doi: 10.1137/0315004. [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), 151-164.doi: 10.1023/A:1004663208905. [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), 365-384.doi: 10.1023/A:1022689517921. [6] M. Ciligot-Travain, On Lagrange Kuhn-Tucker multipliers for Pareto optimization problems, Numerical Functional Analysis and Optimization, 15 (1994), 689-693.doi: 10.1080/01630569408816587. [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), 196-211.doi: 10.1007/s10957-009-9609-1. [8] J. Dutta and V. Veterivel, On approximate minima in vector optimization, Numerical Functional Analysis and Optimization, 22 (2001), 845-859.doi: 10.1081/NFA-100108312. [9] S. Helbig, One New Concept for $\epsilon$-efficency, talk at "Optimization Days 1992", Montreal, 1992. [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), 97-120.doi: 10.1007/s10957-011-9891-6. [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), 483-496.doi: 10.3934/jimo.2011.7.483. [12] D. Gupta and A. Mehra, Two types of approximate saddle points, Numerical Functional Analysis and Optimization,29 (2008), 532-550.doi: 10.1080/01630560802099274. [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), 1046-1058.doi: 10.1016/j.jmaa.2011.12.050. [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), 688-710. [15] C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming, Mathematical Methods of Operations Research, 64 (2006), 165-185. [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), 883-903. [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), 367-383.doi: 10.1007/s10898-004-5904-4. [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), 4331-4346.doi: 10.1016/j.na.2010.02.012. [19] A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces, Optimization, 49 (2001), 137-149.doi: 10.1080/02331930108844524. [20] X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization, Optimization, 51 (2002), 309-321.doi: 10.1080/02331930290019440. [21] Y. W. Huang, Optimality conditions for vector optimization with set-valued maps, Bulletin of the Australian Mathematical Society 66 (2002), 317-330.doi: 10.1017/S0004972700040168. [22] V. Jeyakumar, Convexlike alternative theorems and mathematical programming, Optimization, 16 (1985), 643-652.doi: 10.1080/02331938508843061. [23] S. S. Kutateladze, Convex $\epsilon-$programming, Dokl. Akad. Nauk SSSR, 245 (1979), 1048-1050. [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), 365-375.doi: 10.1023/A:1021786303883. [25] Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps, Journal of Optimization Theory and Applications, 98 (1998), 623-649.doi: 10.1023/A:1022676013609. [26] Z. F. Li and S. Y. Wang, Lagrange multipliers and saddle points in multiobjective programming, Journal of Optimization Theory and Applications, 83 (1994), 63-81.doi: 10.1007/BF02191762. [27] J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency, SIAM Journal on Optimization,19 (2008), 144-162.doi: 10.1137/060676465. [28] J. H. Qiu, A generalized Ekeland vector variational principle and its applications in optimization, Nonlinear Analysis, 71 (2009), 4705-4717.doi: 10.1016/j.na.2009.03.034. [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), 317-327.doi: 10.1016/j.jmaa.2009.06.054. [30] P. H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems, Journal of Optimization Theory and Applications, 119 (2003), 335-356.doi: 10.1023/B:JOTA.0000005449.20614.41. [31] T. Tanaka, A new approach to approximation of solutions in vector optimization problems, In Fushimi, M., Tone ,K.(eds.) Proceedings of APORS, (1994), 497-504. World Scientific, Singapore, 1995. [32] D. J. White, Epsilon efficiency, Journal of Optimization Theory and Applications, 49 (1986), 319-337.doi: 10.1007/BF00940762. [33] X. M. Yang, Alternative theorems and optimality conditions with weakened convexity, Opsearch, 29 (1992), 125-135. [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), 413-427.doi: 10.1023/A:1017535631418. [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), 627-640.doi: 10.1023/A:1026407517917.