Vector nonsmooth variational-like inequalities and optimization problems Original Research Article

In this paper, we first extend the concepts of Clarke’s generalized derivation and subdifferential for a function whose domain is a subset of a metrizable topological vector space. We also introduce and consider a new class of variational-like inequalities, which are called the vector nonsmooth variational-like inequalities. We also establish the relationship between the nonsmooth variational-like inequalities and vector optimization problems under some suitable conditions. In particular, our results extend and generalize the results of Mishra and Wang [S.K. Mishra, S.Y. Wang, Vector variational-like inequalities and nonsmooth vector optimization problems, Nonlinear Anal. 64 (2006), 1939–1945] and Mishra and Noor [S.K. Mishra, M. Aslam Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69–75].