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Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. (English) Zbl 1149.90134

This paper considers a nonsmooth vector optimization problem. The authors use the Hadamard directional derivative for the vector function and a scalarization technique via signed distances to establish necessary and sufficient optimality conditions for weak efficiency as well as firm efficiency. For presenting the necessary first-order conditions, no continuity assumptions are imposed. To derive the second-order sufficient conditions, only the calmness property of the first-order derivatives is assumed. By some examples it is demonstrated that the new conditions are more advantageous than previous results from the literature (see e.g. papers by A. Guerraggio and D. T. Luc, J. Optim. Theory Appl. 109, 615–629 (2001; Zbl 1038.49027), ibid. 116, 117–129 (2003; Zbl 1030.90115) or I. Ginchev, A. Guerraggio, M. Rocca, in: Giannessi, Franco (ed.) et al., Variational analysis and applications. Erice, Italy 2003, New York, NY: Springer. Nonconvex Optimization and its Applications 79, 427–450 (2005; Zbl 1148.90011), Math. Program. 104, No. 2–3 (B), 389–405 (2005; Zbl 1102.90058), Appl. Math., Praha 51, No. 1, 5-36 (2006; Zbl 1164.90399).

MSC:

90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
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