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Optimality and duality for non-Lipschitz multiobjective optimization problems. (English) Zbl 1211.90203
Summary: Nowadays set-valued optimization means set-valued analysis and its application to optimization, and it is an extension of continuous optimization to the set-valued case. In this research area, one investigates optimization problems with constraints and/or an objective function described by set-valued maps, or investigations in set-valued analysis are applied to standard optimization problems. In this article, we are concerned with a set-valued optimization problem \((P)\). Using a notion of approximation derived from Jourani and Thibault, we give necessary and sufficient optimality conditions for \((P)\). Based on necessary optimality conditions given by T. Amahroq and N. Gadhi [J. Glob. Optim. 21, No. 4, 433–441 (2001; Zbl 1175.90409)], our approach consists of formulating the Mond-Weir dual problem \((D)\) and establishing duality theorems for \((P)\) and \((D)\) without any constraint qualification.
MSC:
90C29 Multi-objective and goal programming
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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