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2016, 6(1): 35-44. doi: 10.3934/naco.2016.6.35

## Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps

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

Received  March 2015 Revised  January 2016 Published  January 2016

This paper deals with the characteristics of global proper efficient points and the optimality conditions of vector optimization problems involving generalized convex set-valued maps. Several equivalent properties of global proper efficient points are proposed. Utilizing cone-directed contingent derivative, it presents the unified necessary and sufficient optimality conditions for global proper efficient element in vector optimization problem with cone-arcwise connected set-valued mapping.
Citation: Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35
##### References:
 [1] M. Avriel and I. Zang , Generalized arcwise-connected functions and characterizations of local-global minimum properties, Journal of Optimization Theory and Applications, 32 (1980), 407-425. doi: 10.1007/BF00934030. [2] J. Baier and J. Jahn, On subdifferentials of set-valued maps, Journal of Optimization Theory and Applications, 100 (1980), 233-240. doi: 10.1023/A:1021733402240. [3] H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241. doi: 10.1016/0022-247X(79)90226-9. [4] J. M. Borwein and D. M. Zhuang, Super efficiency in vector optimiation, Transactions of the American Mathematical Society, 338 (1993), 105-122. doi: 10.2307/2154446. [5] Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384. doi: 10.1007/s001860050076. [6] H. W. Corley, Optimality conditions for maximizations of set-valued functions, Journal of Optimization Theory and Applications, 58 (1988), 1-10. doi: 10.1007/BF00939767. [7] C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, Journal of Optimization Theory and Applications, 67 (1990), 297-320. doi: 10.1007/BF00940478. [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), 332-350. doi: 10.1016/S0022-247X(03)00360-3. [9] 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), 12-31. doi: 10.1016/j.jmaa.2004.10.001. [10] M. I. Henig, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Optimization Theory and Applications, 94 (1997), 469-486. [11] J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimzation, Mathematical Methods of Operation Research, 46 (1997), 193-211. doi: 10.1007/BF01217690. [12] C. S. Lalitha, J. Dutta and M. G. Govll, Optimality criteria in set-valued optimization, ournal of the Australian mathematical society, 75 (2003), 221-231. doi: 10.1017/S1446788700003736. [13] D. T. Luc., Contingent derivatives of set-valued maps and applications to vector optimization, Mathematical Programming, 50 (1991), 99-111. doi: 10.1007/BF01594928. [14] X. Q. Yang, Directional derivatives for set-valued mappings and applications, Mathematical Methods of Operations Research, 48 (1998), 273-285. doi: 10.1007/s001860050028. [15] Guolin Yu, Directional derivatives and generalized cone-preinvex set-valued optimizaiton, Acta Mathematica Sinica, Chinese Series (in Chinese), 54 (2011), 875-880. [16] Guolin Yu and Sanyang Liu, Globally proper saddle point in ic-cone-convexlike set-valued optimization problems, Act Mathematica Sinica (English Series), 25 (2009), 1921-1928. doi: 10.1007/s10114-009-6144-9. [17] Guolin Yu and Sanyang Liu, Optimality conditions of globally proper efficient solutions for set-valued optimization problem, Acta Mathematicae Applicatae Sinica(in Chinese), 33 (2010), 150-160. [18] Guolin Yu, Henig globally efficiency for set-valued optimization and vector variational inequality, Journal of Systems Science & Complexity, 27 (2014), 338-349. doi: 10.1007/s11424-014-1215-0. [19] Guolin Yu, Topological properties of Henig globally efficient solutions of set-valued optimization problems, Numerical Algebra, Control and Optimization, 4 (2014), 309-316. doi: 10.3934/naco.2014.4.309. [20] X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 36 (1982), 387-407.

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##### References:
 [1] M. Avriel and I. Zang , Generalized arcwise-connected functions and characterizations of local-global minimum properties, Journal of Optimization Theory and Applications, 32 (1980), 407-425. doi: 10.1007/BF00934030. [2] J. Baier and J. Jahn, On subdifferentials of set-valued maps, Journal of Optimization Theory and Applications, 100 (1980), 233-240. doi: 10.1023/A:1021733402240. [3] H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241. doi: 10.1016/0022-247X(79)90226-9. [4] J. M. Borwein and D. M. Zhuang, Super efficiency in vector optimiation, Transactions of the American Mathematical Society, 338 (1993), 105-122. doi: 10.2307/2154446. [5] Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384. doi: 10.1007/s001860050076. [6] H. W. Corley, Optimality conditions for maximizations of set-valued functions, Journal of Optimization Theory and Applications, 58 (1988), 1-10. doi: 10.1007/BF00939767. [7] C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, Journal of Optimization Theory and Applications, 67 (1990), 297-320. doi: 10.1007/BF00940478. [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), 332-350. doi: 10.1016/S0022-247X(03)00360-3. [9] 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), 12-31. doi: 10.1016/j.jmaa.2004.10.001. [10] M. I. Henig, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Optimization Theory and Applications, 94 (1997), 469-486. [11] J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimzation, Mathematical Methods of Operation Research, 46 (1997), 193-211. doi: 10.1007/BF01217690. [12] C. S. Lalitha, J. Dutta and M. G. Govll, Optimality criteria in set-valued optimization, ournal of the Australian mathematical society, 75 (2003), 221-231. doi: 10.1017/S1446788700003736. [13] D. T. Luc., Contingent derivatives of set-valued maps and applications to vector optimization, Mathematical Programming, 50 (1991), 99-111. doi: 10.1007/BF01594928. [14] X. Q. Yang, Directional derivatives for set-valued mappings and applications, Mathematical Methods of Operations Research, 48 (1998), 273-285. doi: 10.1007/s001860050028. [15] Guolin Yu, Directional derivatives and generalized cone-preinvex set-valued optimizaiton, Acta Mathematica Sinica, Chinese Series (in Chinese), 54 (2011), 875-880. [16] Guolin Yu and Sanyang Liu, Globally proper saddle point in ic-cone-convexlike set-valued optimization problems, Act Mathematica Sinica (English Series), 25 (2009), 1921-1928. doi: 10.1007/s10114-009-6144-9. [17] Guolin Yu and Sanyang Liu, Optimality conditions of globally proper efficient solutions for set-valued optimization problem, Acta Mathematicae Applicatae Sinica(in Chinese), 33 (2010), 150-160. [18] Guolin Yu, Henig globally efficiency for set-valued optimization and vector variational inequality, Journal of Systems Science & Complexity, 27 (2014), 338-349. doi: 10.1007/s11424-014-1215-0. [19] Guolin Yu, Topological properties of Henig globally efficient solutions of set-valued optimization problems, Numerical Algebra, Control and Optimization, 4 (2014), 309-316. doi: 10.3934/naco.2014.4.309. [20] X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 36 (1982), 387-407.
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