×

Superefficiency in vector optimization with nearly subconvexlike set-valued maps. (English) Zbl 1194.90093

Summary: In the framework of locally convex topological vector spaces, we establish a scalarization theorem, a Lagrange multiplier theorem and duality theorems for superefficiency in vector optimization involving nearly subconvexlike set-valued maps.

MSC:

90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
90C46 Optimality conditions and duality in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–491. University of California Press, Berkeley (1951) · Zbl 0044.05903
[2] Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968) · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[3] Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977) · Zbl 0369.90096 · doi:10.1137/0315004
[4] Hartley, R.: On cone efficiency, cone convexity, and cone compactness. SIAM J. Appl. Math. 34, 211–222 (1978) · Zbl 0379.90005 · doi:10.1137/0134018
[5] Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979) · Zbl 0418.90081 · doi:10.1016/0022-247X(79)90226-9
[6] Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982) · Zbl 0452.90073 · doi:10.1007/BF00934353
[7] Borwein, J.M., Zhuang, D.: Superefficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993) · Zbl 0796.90045 · doi:10.2307/2154446
[8] Zheng, X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997) · Zbl 0889.90141 · doi:10.1023/A:1022648115446
[9] Corley, H.W.: Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54, 489–500 (1987) · Zbl 0595.90085 · doi:10.1007/BF00940198
[10] Lin, L.J.: Optimization of set-valued functions. J. Math. Anal. Appl. 186, 30–51 (1994) · Zbl 0987.49011 · doi:10.1006/jmaa.1994.1284
[11] Gong, X.H.: Connectedness of efficient solution sets for set-valued maps in normed spaces. J. Optim. Theory Appl. 83, 83–96 (1994) · Zbl 0845.90104 · doi:10.1007/BF02191763
[12] Li, Z.F., Chen, G.Y.: Lagrange multipliers, saddle points and duality in vector optimization of set-valued maps. J. Math. Anal. Appl. 215, 297–316 (1997) · Zbl 0893.90150 · doi:10.1006/jmaa.1997.5568
[13] Song, W.: Lagrangian duality for minimization of nonconvex multifunctions. J. Optim. Theory Appl. 93, 167–182 (1997) · Zbl 0901.90161 · doi:10.1023/A:1022658019642
[14] Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 1187–1200 (1998) · Zbl 0927.90095
[15] Li, Z.F.: Benson proper efficiency in vector optimization of set-valued maps. J. Optim. Theory Appl. 98, 623–649 (1998) · Zbl 0913.90235 · doi:10.1023/A:1022676013609
[16] Rong, W.D., Wu, Y.N.: Characterizations of superefficiency in cone-convexlike vector optimization with set-valued maps. Math. Methods Oper. Res. 48, 247–258 (1998) · Zbl 0930.90078 · doi:10.1007/s001860050026
[17] Mehra, A.: Superefficiency in vector optimization with nearly convexlike set-valued maps. J. Math. Anal. Appl. 276, 815–832 (2002) · Zbl 1106.90375 · doi:10.1016/S0022-247X(02)00452-3
[18] Cristescu, R.: Ordered Vector Spaces and Linear Operators. Abacus, Kent (1976) · Zbl 0322.46010
[19] Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1971) · Zbl 0212.14001
[20] Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001) · Zbl 1012.90061 · doi:10.1023/A:1017535631418
[21] Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119, 335–356 (2003) · Zbl 1055.90066 · doi:10.1023/B:JOTA.0000005449.20614.41
[22] Cheng, Y.H., Fu, W.T.: Strong efficiency in a locally convex space. Math. Methods Oper. Res. 50, 373–384 (1999) · Zbl 0947.90104 · doi:10.1007/s001860050076
[23] Zhuang, D.: Density results for proper efficiencies. SIAM J. Control Optim. 32, 51–58 (1994) · Zbl 0798.49028 · doi:10.1137/S0363012989171518
[24] Gong, X.H., Dong, H.B., Wang, S.Y.: Optimality conditions for proper efficient solutions of vector set-valued optimization. J. Math. Anal. Appl. 284, 332–350 (2003) · Zbl 1160.90649 · doi:10.1016/S0022-247X(03)00360-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.