Characterizations of nonconvex optimization problems via variational inequalities

In this paper, we deal with two problems from the theory of nonconvex nonsmooth analysis; The characterization of nonsmooth quasiconvex functions, and connections between nonsmooth constraint optimization problems via variational inequalities. For the first problem, we provide different characterizations for nonsmooth quasiconvex functions, while for the second problem, a full connection between constraint optimization problems and Stampacchia and Minty variational inequalities is provided, in both cases, neither differentiability nor convexity nor continuity assumptions are considered. As a corollary, we recover well-known results from convex analysis.