×

First order optimality conditions in vector optimization involving stable functions. (English) Zbl 1191.90056

Summary: We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.

MSC:

90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
49K27 Optimality conditions for problems in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubin JP, Set-Valued Analysis (1990)
[2] DOI: 10.1287/moor.1.2.165 · Zbl 0404.90100 · doi:10.1287/moor.1.2.165
[3] Clarke FH, Optimization and Nonsmooth Analysis (1983)
[4] DOI: 10.1080/01630568908816290 · Zbl 0645.90076 · doi:10.1080/01630568908816290
[5] DOI: 10.1051/ro:2004023 · Zbl 1158.90421 · doi:10.1051/ro:2004023
[6] DOI: 10.1023/A:1017519922669 · Zbl 1038.49027 · doi:10.1023/A:1017519922669
[7] Hestenes MR, Optimization theory. The finite dimensional case (1981)
[8] DOI: 10.1007/BF01442544 · Zbl 0389.90088 · doi:10.1007/BF01442544
[9] DOI: 10.1006/jmaa.2001.7588 · Zbl 1010.90075 · doi:10.1006/jmaa.2001.7588
[10] DOI: 10.1016/S0022-247X(02)00064-1 · Zbl 1008.90051 · doi:10.1016/S0022-247X(02)00064-1
[11] DOI: 10.1081/NFA-120006695 · Zbl 1025.90023 · doi:10.1081/NFA-120006695
[12] DOI: 10.1016/S0022-247X(03)00337-8 · Zbl 1033.90120 · doi:10.1016/S0022-247X(03)00337-8
[13] DOI: 10.1007/s10957-006-9158-9 · Zbl 1149.90176 · doi:10.1007/s10957-006-9158-9
[14] DOI: 10.1023/A:1022272728208 · Zbl 0995.90085 · doi:10.1023/A:1022272728208
[15] DOI: 10.1007/BF01594928 · Zbl 0718.90080 · doi:10.1007/BF01594928
[16] Luc DT, Vector optimization. Lecture delivered at the summer school ’‘Generalized Convexity and Monotonicity” (1999)
[17] Michel P, Diff. Integ. Eq 5 pp 433– (1992)
[18] DOI: 10.1080/02331930108844561 · Zbl 1039.90069 · doi:10.1080/02331930108844561
[19] DOI: 10.1080/02331939808844391 · Zbl 0935.49009 · doi:10.1080/02331939808844391
[20] DOI: 10.1007/978-3-642-02431-3 · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3
[21] DOI: 10.1007/BF00938820 · Zbl 0593.90071 · doi:10.1007/BF00938820
[22] DOI: 10.1137/0324061 · Zbl 0604.49017 · doi:10.1137/0324061
[23] DOI: 10.1007/s10107-004-0569-9 · Zbl 1099.49020 · doi:10.1007/s10107-004-0569-9
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.