×

Lagrangian function and duality theory in multiobjective programming with set functions. (English) Zbl 0619.90072

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions, D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex cone D. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.

MSC:

90C31 Sensitivity, stability, parametric optimization
49N15 Duality theory (optimization)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319-377, 1974. · Zbl 0268.90057 · doi:10.1007/BF00932614
[2] Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509-529, 1979. · Zbl 0378.90100 · doi:10.1007/BF00933437
[3] Chou, J. H., Hsia, W. S., andLee, T. Y.,On Multiple Objective Programming Problems with Set Functions, Journal of Mathematical Analysis and Applications, Vol. 15, pp. 383-394, 1985. · Zbl 0564.90069 · doi:10.1016/0022-247X(85)90055-1
[4] Hsia, W. S., andLee, T. Y.,Proper D-Solutions of Multiobjective Programming Problems with Set Functions, Journal of Optimization Theory and Applications, Vol. 53, pp. 247-258, 1987. · Zbl 0595.90084 · doi:10.1007/BF00939217
[5] Geoffrion, A. M.,Proper Efficiency and 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
[6] Morris, R. J. T.,Optimal Constrained Selection of Measurable Subset, Journal of Mathematical Analysis and Applications, Vol. 70, pp. 546-562, 1979. · Zbl 0417.49032 · doi:10.1016/0022-247X(79)90064-7
[7] Chou, J. H., Hsia, W. S., andLee, T. Y.,Epigraphs of Convex Set Functions, Journal of Mathematical Analysis and Applications, Vol. 118, pp. 247-254, 1986. · Zbl 0599.49014 · doi:10.1016/0022-247X(86)90306-9
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.