In vector optimization problems several notions of proper efficiency have been proposed, in order to rule out some undesirable situations (tolerated by the definition of efficiency). The relationships between the above different definitions have been studied by many authors. In this paper we again show these relationships, under the assumption that the involved functions are not differentiable but only Lipschitzian. The properties of Clarke’s generalized derivative and subdifferential are here used, in order to get a picture similar to the one obtained for the differentiable case. Then, by means of suitable generalized convexity assumptions, we obtain some equivalences of the different notions of proper efficiency considered, with respect to the notion of efficiency. The results of this paper may be viewed as a generalization of the situation established for the smooth case.
Proper efficiency and generalized convexity in nonsmooth vector optimization problems
GUERRAGGIO, ANGELO
2001-01-01
Abstract
In vector optimization problems several notions of proper efficiency have been proposed, in order to rule out some undesirable situations (tolerated by the definition of efficiency). The relationships between the above different definitions have been studied by many authors. In this paper we again show these relationships, under the assumption that the involved functions are not differentiable but only Lipschitzian. The properties of Clarke’s generalized derivative and subdifferential are here used, in order to get a picture similar to the one obtained for the differentiable case. Then, by means of suitable generalized convexity assumptions, we obtain some equivalences of the different notions of proper efficiency considered, with respect to the notion of efficiency. The results of this paper may be viewed as a generalization of the situation established for the smooth case.
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