Abstract
The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.
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Communicated by G. Leitmann
The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.
The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.
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Benson, H.P. Existence of efficient solutions for vector maximization problems. J Optim Theory Appl 26, 569–580 (1978). https://doi.org/10.1007/BF00933152
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DOI: https://doi.org/10.1007/BF00933152