Nonsmooth quasi-variational-like inequalities with applications to nonsmooth vector quasi-optimization

We consider nonsmooth vector quasi-variational-like inequalities and nonsmooth vector quasioptimization 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.