Chen, G. Y.; Rong, W. D. Characterizations of the Benson proper efficiency for nonconvex vector optimization. (English) Zbl 0908.90225 J. Optimization Theory Appl. 98, No. 2, 365-384 (1998). Summary: Under generalized cone-subconvexlikeness for vector-valued mappings in locally-convex Hausdorff topological vector spaces, a Gordan-form alternative theorem is derived. Some characterizations of the Benson proper efficiency under this general convexity are established in terms of scalarization, Lagrangian multipliers, saddle-point criterion, and duality. Cited in 1 ReviewCited in 32 Documents MSC: 90C29 Multi-objective and goal programming 90C48 Programming in abstract spaces Keywords:generalized cone-subconvexlikeness; vector optimization; proper efficiency; scalarization; Lagrangian multipliers; saddle-point criterion; duality PDFBibTeX XMLCite \textit{G. Y. Chen} and \textit{W. D. Rong}, J. Optim. Theory Appl. 98, No. 2, 365--384 (1998; Zbl 0908.90225) Full Text: DOI References: [1] Kuhn, H. W., and Tucker, A. W., Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, pp. 481-492, 1951. [2] Geoffrion, A. M., Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618-630, 1968. · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1 [3] Hartley, R., On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211-222, 1978. · Zbl 0379.90005 · doi:10.1137/0134018 [4] Tamura, K., and Arai, S., On Proper and Improper Efficient Solutions of Optimal Problems with Multicriteria, Journal of Optimization Theory and Applications, Vol. 38, pp. 191-205, 1982. · Zbl 0471.49003 · doi:10.1007/BF00934082 [5] Borwein, J. M., Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57-63, 1977. · Zbl 0369.90096 · doi:10.1137/0315004 [6] Benson, H. P., An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 232-241, 1979. · Zbl 0418.90081 · doi:10.1016/0022-247X(79)90226-9 [7] Henig, M. I., Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387-407, 1982. · Zbl 0452.90073 · doi:10.1007/BF00934353 [8] Jahn, J., A Characterization of Properly Minimal Elements of a Set, SIAM Journal on Control and Optimization, Vol. 23, pp. 649-656, 1985. · Zbl 0579.90086 · doi:10.1137/0323041 [9] Dauer, J. P., and Gallagher, R. J., Positive Proper Efficiency and Related Cone Results in Vector Optimization Theory, SIAM Journal on Control and Optimization, Vol. 28, pp. 158-172, 1990. · Zbl 0697.90072 · doi:10.1137/0328008 [10] Borwein, J. M., and Zhuang, D. M., Superefficiency in Convex Vector Optimization, ZOR-Mathematical Methods of Operations Research, Vol. 35, pp. 175-184, 1991. · Zbl 0777.90050 · doi:10.1007/BF01415905 [11] Borwein, J. M., and Zhuang, D. M., Superefficiency in Vector Optimization, Transactions of the American Mathematical Society, Vol. 38, pp. 105-122, 1993. · Zbl 0796.90045 · doi:10.1090/S0002-9947-1993-1098432-5 [12] Jeyakumar, V., A Generalization of a Minimax Theorem of Fan via a Theorem of the Alternative, Journal of Optimization Theory and Applications, Vol. 48, pp. 525-533, 1986. · Zbl 0563.49006 · doi:10.1007/BF00940575 [13] Guerraggio, A., Molho, E., and Zaffaroni, A., On the Notion of Proper Efficiency in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 82, pp. 1-21, 1994. · Zbl 0827.90123 · doi:10.1007/BF02191776 [14] Li, Z. F., and Wang, S. Y., Lagrange Multipliers and Saddle Points in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 83, pp. 63-81, 1994. · Zbl 0823.90107 · doi:10.1007/BF02191762 [15] Yang, X. M., Alternative Theorems and Optimality Conditions with Weakened Convexity, Opsearch, Vol. 29, pp. 115-125, 1992. · Zbl 0757.90075 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.