  Since the seminal papers by Giannessi, an interesting topic in vector
optimization has been the characterization of (weak) efficiency thorough Minty
and Stampacchia type variational inequalities. Several results have been proved
to extend those known for the scalar case. However, in order to introduce a
proper definition of variational inequality, some assumptions are usually made
that may eventually be questioned. We find two major drawbacks in the papers we
considered, that arise when defining generalized derivatives for vector-valued
functions. First, some authors introduce set-valued derivatives for
single-valued problems, thus completely changing the setting of the problem.
Second, when dealing with Dini-type derivatives, infinite elements may occurs.
The approach to handle this problem is not yet uniquely defined in the
literature, therefore, when considered, the definition proposed may seem
arbitrary. Indeed these problems are strictly related with the lack of a
complete order in the image space of a vector-valued function. We propose an
alternative approach to study vector optimization, by considering an set-valued
counterpart defined with values in a conlinear space. The structure of this
space allows to overcome the previous difficulties and to obtain variational
inequality characterization of weak efficiency as a straightforward application
of scalar arguments.
