Directional Subdifferentials at Infinity and Its Applications

Abstract: This paper investigates the behavior of sets and functions at infinity by introducing new concepts, namely directional normal cones at infinity for unbounded sets, along with limiting and singular subdifferentials at infinity in the direction for extended real-valued functions. We develop several calculus rules for these concepts and then apply them to nonsmooth optimization problems. The applications include establishing directional optimality conditions at infinity, analyzing the coercivity, proving the compactness of the global solution set, and examining properties such as weak sharp minima and error bounds at infinity. To demonstrate the effectiveness of the proposed approach, illustrative examples are provided and compared with existing results.