Clarke’s Tangent Cones, Subgradients, Optimality Conditions, and the Lipschitzness at Infinity

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1732-1754, June 2024.
Abstract. We first study Clarke’s tangent cones at infinity to unbounded subsets of [math]. 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 [math] 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 semicontinuous functions.