# 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. [2] S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71. [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. [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. [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. [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. [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. [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. [9] S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (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. 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. 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. 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. 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. [15] C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165. [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. [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. [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. [19] A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137. doi: 10.1080/02331930108844524. [20] X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309. 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), 66 (2002), 317. doi: 10.1017/S0004972700040168. [22] V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643. doi: 10.1080/02331938508843061. [23] S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048. [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. [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. [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. [27] J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144. doi: 10.1137/060676465. [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. [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. [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. [31] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497. [32] D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319. doi: 10.1007/BF00940762. [33] X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125. [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. [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.

show all references

##### References:
 [1] T. Amahroq and A. Taa, On Lagrange Kuhn-Tucker multipliers for multiobjective optimization problems,, Optimization, 41 (1997), 159. doi: 10.1080/02331939708844332. [2] S. Bolintineanu, Vector variational principles: $\epsilon$-efficiency and scalar stationarity,, Journal of Convex Analysis, 8 (2001), 71. [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. [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. [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. [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. [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. [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. [9] S. Helbig, One New Concept for $\epsilon$-efficency,, talk at, (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. 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. 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. 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. 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. [15] C. Gutiérrez, B. Jiménez and V. Novo, On approximate efficiency in multiobjective programming,, Mathematical Methods of Operations Research, 64 (2006), 165. [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. [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. [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. [19] A. Hamel, An $\epsilon$-Lagrange multiplier rule for a mathematical programming problem on Banach spaces,, Optimization, 49 (2001), 137. doi: 10.1080/02331930108844524. [20] X. X. Huang, Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization,, Optimization, 51 (2002), 309. 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), 66 (2002), 317. doi: 10.1017/S0004972700040168. [22] V. Jeyakumar, Convexlike alternative theorems and mathematical programming,, Optimization, 16 (1985), 643. doi: 10.1080/02331938508843061. [23] S. S. Kutateladze, Convex $\epsilon-$programming,, Dokl. Akad. Nauk SSSR, 245 (1979), 1048. [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. [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. [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. [27] J. H. Qiu, Dual characterization and scalarization for Benson properly efficiency,, SIAM Journal on Optimization, 19 (2008), 144. doi: 10.1137/060676465. [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. [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. [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. [31] T. Tanaka, A new approach to approximation of solutions in vector optimization problems,, In Fushimi, (1994), 497. [32] D. J. White, Epsilon efficiency,, Journal of Optimization Theory and Applications, 49 (1986), 319. doi: 10.1007/BF00940762. [33] X. M. Yang, Alternative theorems and optimality conditions with weakened convexity,, Opsearch, 29 (1992), 125. [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. [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.
 [1] 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 [2] Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35 [3] Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1 [4] Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 [5] Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309 [6] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [7] 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 [8] Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031 [9] Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051 [10] Yu Han, Nan-Jing Huang. Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1135-1151. doi: 10.3934/jimo.2016.12.1135 [11] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 [12] Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013 [13] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 [14] Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57 [15] Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019 [16] Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 [17] Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087 [18] Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303 [19] Hsien-Chung Wu. Solving the interval-valued optimization problems based on the concept of null set. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1157-1178. doi: 10.3934/jimo.2018004 [20] Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

2018 Impact Factor: 1.025

## Metrics

• PDF downloads (12)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]