About the duality gap in vector optimization

Giancarlo Bigi and Massimo Pappalardo


Duality is one of the most important topics in optimization both from a theoretical and algorithmic point of view. In scalar optimization, one generally looks for a dual problem in such a way that the difference between the optimal values is non-negative, small and possibly zero. This difference is called duality gap. However, such a definition cannot be applied to vector optimization easily, since a vector program has not just an optimal value but a set of optimal ones. A first attempt to analyse the vector case appeared in the seventies but it was based on scalarization techniques and the considered duality gap was between scalar problems. Though a large number of papers dealing with duality for vector optimization have been published, to the best of our knowledge, studies about the duality gap have not been carried out. The first difficulty to overcome is the definition of duality gap itself since the analysis requires at least the comparison between two sets of vector optimal values.The aim of this note is to address possible answers to this question and not to present new duality results. Therefore, we review some of the vector optimization duality schemes developed in literature and we propose some concepts of duality gap, which match the known results. Some numerical examples are also provided.

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