# 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

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.  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
 [1] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [2] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [3] Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 [4] Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 [5] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [6] Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 [7] Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 [8] Chaoqian Li, Yajun Liu, Yaotang Li. Note on $Z$-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 [9] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 [10] A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 [11] Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 [12] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 [13] Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 [14] Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 [15] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [16] Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 [17] Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151 [18] Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 [19] M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072 [20] Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

2019 Impact Factor: 1.366