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2014, 4(4): 309-316. doi: 10.3934/naco.2014.4.309

## Topological properties of Henig globally efficient solutions of set-valued problems

 1 Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan 750021, China

Received  March 2013 Revised  October 2014 Published  December 2014

This paper deals with the topological properties of Henig globally efficiency in vector optimization problem with set-valued mapping. The closednesst and compactness results are presented for Henig globally efficiency, Especially, under the assumption of ic-cone-convexlikeness, the connectedness of Henig globally efficient set is obtained. As the application of the results, the relevant topological properties of Benson proper efficiency are also presented.
Citation: 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
##### References:
 [1] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis., Wiley, (1984). doi: 10.1007/978-1-4612-0873-0. [2] H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, Journal of Mathematical Analysis and Applications, 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9. [3] J. M. Borwein and D. M. Zhuang, Super efficiency in vector optimization,, Trans. Amer. Math. Soc., 338 (1993), 105. doi: 10.2307/2154446. [4] H. W. Corley, Optimality conditions for maximizations of set-valued functions,, Journal of Optimization Theory and Applications, 54 (1987), 489. doi: 10.1007/BF00940198. [5] X. H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior,, Journal of Mathematical Analysis and Applications, 307 (2005), 12. doi: 10.1016/j.jmaa.2004.10.001. [6] X. H. Gong, Connectedness of super efficient solution sets for set-valued maps in Banach spaces,, Mathematical Methods of Operation Research, 44 (1996), 135. doi: 10.1007/BF01246333. [7] X. H. Gong, Connectedness of efficient solution sets for set-valued maps in normed spaces,, Journal of Optimization Theory and Applications, 83 (1994), 83. doi: 10.1007/BF02191763. [8] X. H. Gong, H. B. Dong and S. Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, Journal of Mathematical Analysis and Applications, 284 (2003), 332. doi: 10.1016/S0022-247X(03)00360-3. [9] M. I. Henig, Proper efficiency with respect to cones,, Journal of Optimization Theory and Applications, 36 (1982), 387. doi: 10.1007/BF00934353. [10] J. B. Hiriart-Urruty, Images of connected sets by semicontinuous multifunctions,, Journal of Mathematical Analysis and Applications, 111 (1985), 407. doi: 10.1016/0022-247X(85)90225-2. [11] 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. [12] Y. X. Li, Topological structure of efficient set of optimization problem so set-valued mapping,, Chinese Annals of Mathematics, 15B (1994), 115. [13] Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of set-valued maps,, Mathematical Methods of Operation Research, 48 (1998), 207. doi: 10.1007/s001860050023. [14] Z. F. Li and G, Y, Chen, Lagrangian multipliers, saddle points and duality in vector optimizaition of set-valued maps,, Journal of Mathematical Analysis and Applications, 215 (1997), 297. doi: 10.1006/jmaa.1997.5568. [15] P. H. Naccache, Connectedness of the set of nondominated outcomes in multicriteria optimization,, Journal of Optimization Theory and Applications, 25 (1978), 459. doi: 10.1007/BF00932907. [16] Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for set-valued optimization problems,, \emph{Journal of Optimization Theory and Applications}, 152 (2012), 439. doi: 10.1007/s10957-011-9906-3. [17] P. H. Sach, New generalized convexity notion for set-valued maps and application to vector optimization,, Journal of Optimization Theory and Applications, 125 (2005), 157. doi: 10.1007/s10957-004-1716-4. [18] W. Song, Lagrangian duality for minimization of nonconvex multifunctions,, Journal of Optimization Theory and Applications, 93 (1997), 167. doi: 10.1023/A:1022658019642. [19] A. R. Warburton, Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives,, Journal of Optimization Theory and Applications, 140 (1983), 537. doi: 10.1007/BF00933970. [20] Yihong Xu and Xiaoshuai Song, The relationship between ic-cone-convexness and nearly cone-subconvexlikeness,, Applied Mathematics Letters, 24 (2011), 1622. doi: 10.1016/j.aml.2011.04.018. [21] X. M. Yang, X. Q. Yang and G. Y. Cheng, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627. doi: 10.1023/A:1004613630675. [22] Guolin Yu and Sanyang Liu, Optimality conditions of globally proper efficient solutions for set-valued optimization problem,, Acta Mathematicae Applicatae Sinica, 33 (2010), 150. [23] Guolin Yu and Sanyang Liu, Globally proper saddle point in ic-cone-convexlike set-valued optimization problems,, Act Mathematica Sinica (English Series), 25 (2009), 1921. doi: 10.1007/s10114-009-6144-9.

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##### References:
 [1] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis., Wiley, (1984). doi: 10.1007/978-1-4612-0873-0. [2] H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, Journal of Mathematical Analysis and Applications, 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9. [3] J. M. Borwein and D. M. Zhuang, Super efficiency in vector optimization,, Trans. Amer. Math. Soc., 338 (1993), 105. doi: 10.2307/2154446. [4] H. W. Corley, Optimality conditions for maximizations of set-valued functions,, Journal of Optimization Theory and Applications, 54 (1987), 489. doi: 10.1007/BF00940198. [5] X. H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior,, Journal of Mathematical Analysis and Applications, 307 (2005), 12. doi: 10.1016/j.jmaa.2004.10.001. [6] X. H. Gong, Connectedness of super efficient solution sets for set-valued maps in Banach spaces,, Mathematical Methods of Operation Research, 44 (1996), 135. doi: 10.1007/BF01246333. [7] X. H. Gong, Connectedness of efficient solution sets for set-valued maps in normed spaces,, Journal of Optimization Theory and Applications, 83 (1994), 83. doi: 10.1007/BF02191763. [8] X. H. Gong, H. B. Dong and S. Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, Journal of Mathematical Analysis and Applications, 284 (2003), 332. doi: 10.1016/S0022-247X(03)00360-3. [9] M. I. Henig, Proper efficiency with respect to cones,, Journal of Optimization Theory and Applications, 36 (1982), 387. doi: 10.1007/BF00934353. [10] J. B. Hiriart-Urruty, Images of connected sets by semicontinuous multifunctions,, Journal of Mathematical Analysis and Applications, 111 (1985), 407. doi: 10.1016/0022-247X(85)90225-2. [11] 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. [12] Y. X. Li, Topological structure of efficient set of optimization problem so set-valued mapping,, Chinese Annals of Mathematics, 15B (1994), 115. [13] Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of set-valued maps,, Mathematical Methods of Operation Research, 48 (1998), 207. doi: 10.1007/s001860050023. [14] Z. F. Li and G, Y, Chen, Lagrangian multipliers, saddle points and duality in vector optimizaition of set-valued maps,, Journal of Mathematical Analysis and Applications, 215 (1997), 297. doi: 10.1006/jmaa.1997.5568. [15] P. H. Naccache, Connectedness of the set of nondominated outcomes in multicriteria optimization,, Journal of Optimization Theory and Applications, 25 (1978), 459. doi: 10.1007/BF00932907. [16] Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for set-valued optimization problems,, \emph{Journal of Optimization Theory and Applications}, 152 (2012), 439. doi: 10.1007/s10957-011-9906-3. [17] P. H. Sach, New generalized convexity notion for set-valued maps and application to vector optimization,, Journal of Optimization Theory and Applications, 125 (2005), 157. doi: 10.1007/s10957-004-1716-4. [18] W. Song, Lagrangian duality for minimization of nonconvex multifunctions,, Journal of Optimization Theory and Applications, 93 (1997), 167. doi: 10.1023/A:1022658019642. [19] A. R. Warburton, Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives,, Journal of Optimization Theory and Applications, 140 (1983), 537. doi: 10.1007/BF00933970. [20] Yihong Xu and Xiaoshuai Song, The relationship between ic-cone-convexness and nearly cone-subconvexlikeness,, Applied Mathematics Letters, 24 (2011), 1622. doi: 10.1016/j.aml.2011.04.018. [21] X. M. Yang, X. Q. Yang and G. Y. Cheng, Theorems of the alternative and optimization with set-valued maps,, Journal of Optimization Theory and Applications, 107 (2000), 627. doi: 10.1023/A:1004613630675. [22] Guolin Yu and Sanyang Liu, Optimality conditions of globally proper efficient solutions for set-valued optimization problem,, Acta Mathematicae Applicatae Sinica, 33 (2010), 150. [23] Guolin Yu and Sanyang Liu, Globally proper saddle point in ic-cone-convexlike set-valued optimization problems,, Act Mathematica Sinica (English Series), 25 (2009), 1921. doi: 10.1007/s10114-009-6144-9.
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