In this work, we extend the relationships between convex functions and corresponding monotone maps to pseudo-convexity and the corresponding pseudo-monotonicity of their sub-differentiable maps. We characterize the lower semi-continuous Clarke-Rockafeller sub-differentiable pseudo-convex functions  on convex subset  of infinite-dimensional real Banach spaces  with respect to the corresponding monotonicity of their Clarke-Rockafeller sub-differential operators  and obtain the optimality conditions, using variational inequalities.