Super efficiency in vector optimization with nearly convexlike set-valued maps. (English) Zbl 1106.90375
Summary: In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency.
MSC:
| 90C29 | Multi-objective and goal programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
References:
| [1] | 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 |
| [2] | Borwein, J. M., Proper efficient points for maximization with respect to cones, SIAM J. Control Optim., 15, 57-63 (1977) · Zbl 0369.90096 |
| [3] | Borwein, J. M.; Zhuang, D. M., Super efficiency in convex vector optimization, Math. Methods Oper. Res., 35, 175-184 (1991) · Zbl 0777.90050 |
| [4] | Borwein, J. M.; Zhuang, D. M., Super efficiency in vector optimization, Trans. Amer. Math. Soc., 338, 105-122 (1993) · Zbl 0796.90045 |
| [5] | Corley, H. W., Existence and Lagrangian duality for maximization of set-valued function, J. Optim. Theory Appl., 54, 489-500 (1987) · Zbl 0595.90085 |
| [6] | Corley, H. W., Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl., 58, 1-10 (1988) · Zbl 0956.90509 |
| [7] | Geoffrion, A. M., Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22, 618-630 (1968) · Zbl 0181.22806 |
| [8] | Jahn, J., Mathematical Vector Optimization in Partially Ordered Linear Spaces (1986), Peter Lang: Peter Lang Frankfurt am Main · Zbl 0578.90048 |
| [9] | Li, Z. F., Benson proper efficiency in the vector optimization of set-valued maps, J. Optim. Theory Appl., 98, 623-649 (1998) · Zbl 0913.90235 |
| [10] | 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 |
| [11] | Lin, L. J., Optimization of set-valued functions, J. Math. Anal. Appl., 186, 30-51 (1994) · Zbl 0987.49011 |
| [12] | Rong, W. D.; Wu, Y. N., Characterization of super efficiency in cone-convexlike vector optimization with set-valued maps, Math. Methods Oper. Res., 48, 247-258 (1998) · Zbl 0930.90078 |
| [13] | Song, W., Lagrangian duality for minimization of nonconvex multifunctions, J. Optim. Theory Appl., 93, 167-182 (1997) · Zbl 0901.90161 |
| [14] | Yu, P. L., Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjective, J. Optim. Theory Appl., 14, 319-377 (1974) · Zbl 0268.90057 |
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.
