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A note on “Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality”. (English) Zbl 1343.32007

Summary: The note points out that the sufficiency of proposition 2.1 in [N. Le H. Anh, ibid. 18, No. 3, 449–473 (2014; Zbl 1298.49023)] is erroneous and we provide an example to illustrate it. Also the proof of proposition 2.2 in [loc. cit.] is incorrect and we give a new proof.

MSC:

32F17 Other notions of convexity in relation to several complex variables
46G05 Derivatives of functions in infinite-dimensional spaces
90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming

Citations:

Zbl 1298.49023
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References:

[1] Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107, 627-640 (2000) · Zbl 0966.90071 · doi:10.1023/A:1026407517917
[2] 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
[3] Sach, P.H.: New generalized convexity notion for set-valued maps and application to vector optimization. J. Optim. Theory Appl. 125, 157-179 (2005) · Zbl 1090.90174 · doi:10.1007/s10957-004-1716-4
[4] Xu, Yihong: Song, Xiaoshuai: Relationship between ic-cone-convexness and nearly cone-subconvexlikeness. Appl. Math. Lett. 24, 1622-1624 (2011) · Zbl 1218.52003 · doi:10.1016/j.aml.2011.04.018
[5] Anh, N.L.H.: Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality. Positivity 18, 449-473 (2014) · Zbl 1298.49023 · doi:10.1007/s11117-013-0254-4
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