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Duality in vector optimization. II: Vector quasiconcave programming. (English) Zbl 0575.49006

[For part I see ibid. 20, 304-313 (1984; Zbl 0556.49010).]
On the basis of the abstract theory presented in the first part, a duality theory is developed for the vector quasiconcave programming. In Section 3 some necessary concepts and assertions of (quasi)convexity are introduced. Section 4 deals with the duality theory in vector quasiconcave programming with affine constraints. Finally, in Section 5 a limit approach is proposed to define the dual problems for the vector quasiconcave programming with convex constraints. The third part of this paper is described in the following review.

MSC:

49N15 Duality theory (optimization)
49K27 Optimality conditions for problems in abstract spaces
90C48 Programming in abstract spaces
47H99 Nonlinear operators and their properties
49J45 Methods involving semicontinuity and convergence; relaxation
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
90C25 Convex programming
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References:

[1] Tran Quoc Chien: Duality in vector optimization. Part 1: Abstract duality scheme. Kybernetika 20 (1984), 3, 304-313. · Zbl 0556.49010
[2] B. Martos: Nonlinear Programming: Theory and Methods. Akadémiai Kiadó, Budapest 1975. · Zbl 0357.90027
[3] G. S. Rubinstein: Separation theorems of convex sets. Sibirsk. Mat. Ž. 5 (1964), 5, 1098-1124. In Russian. · Zbl 0137.15202
[4] V. G. Demjanov, L. V. Vasiljev: Nonsmooth Optimization. Nauka, Moskva 1981. In Russian.
[5] R. Holmes: Geometrical Functional Analysis and Its Applications. Springer-Verlag, Berlin-Heidelberg-New York 1975. · Zbl 0336.46001
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