There are various theoretical, algorithmic, and practical reasons for developing necessary and sufficient conditions for the efficient set of a multiple objective mathematical programming problem (P) to be closed. Yet only a small number of results of this type, limited to special cases of Problem (P), have been developed. In this article we present a necessary condition and several sufficient conditions for the closedness of the efficient sets of more general cases of Problem (P) than have heretofore been studied. Our approach relies in part upon generalizing the concepts of quasi-concavity and strict quasi-concavity for real-valued functions to vector-valued functions. Our approach also relies upon some new characterizations of an efficient solution for Problem (P) that we develop in the article.
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