Applying Set Optimization to Weak Efficiency

Abstract

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.