Clarke's tangent cones, subgradients, optimality conditions and the Lipschitzness at infinity

We first study Clarke’s tangent cones at infinity to unbounded subsets of ${\mathbb{R}}^n\mathrm{.}$ We prove that these cones are closed convex and show a characterization of their interiors. We then study subgradients at infinity for extended real value functions on ${\mathbb{R}}^n$ and derive necessary optimality conditions at infinity for optimization problems. We also give a number of rules for the computing of subgradients at infinity and provide some characterizations of the Lipschitz continuity at infinity for lower semi-continuous functions.