About the duality gap in vector optimization
Giancarlo Bigi and Massimo Pappalardo
Summary
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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