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The usual approximation solutions with variable ordering structures fail in some cases such as Examples 2.1 and 2.2, where the intersection of all ordering cones equals to the zero element. Inspired by this, we introduce a new approximate nondominated solution depending on a variable point substituting for a fixed point, which generalize the usual solutions. By proposing two types of separation bi-functionals, optimality characterizations in a unified way are concluded for various approximate nondominated solutions. Augmented dual cones and max scalarizing functional are proved to associate closely with some specific separation bi-functionals. Considering only weakly nondominated or nondominated minimizers, we also derive the Fermat rules based on the tools of variational analysis. Particularly, the important setting for variable ordering structures equipped with Bishop-Phelps cones are examined based on our theories.

Optimality characterizations for approximate nondominated and Fermat rules for nondominated solutions