Nonsmooth Quasi-Variational-like Inequalities with Applications to Nonsmooth Vector Quasi-Optimization

Abstract

We consider nonsmooth vector quasi-variational-like inequalities and
nonsmooth vector quasi-optimization problems. We utilize the method of scalarization
to define
nonsmooth quasi-variational-like inequalities by means of Clarke generalized
directional derivative. We then
study their relations with the problem of vector quasi-optimization and its scalarized
version.
Under the assumption of pseudomonotonicity and then densely pseudomonotonicity,
we present some existence results for solutions to nonsmooth quasi-variational-like
inequalities. To the best of our knowledge, the results we obtained are new in the
sense of utilizing the scalarization method.