Directional Subdifferentials at Infinity and Its Applications
Le Ngoc Kien, Nguyen Van Tuyen, Tran Van Nghi
TL;DR
This work develops a directional framework for subdifferentials and normal cones at infinity, introducing the directional normal cone at infinity $N_{\Omega}(\infty; u)$ and directional limiting/singular subdifferentials $\partial f(\infty; u)$, $\partial^{\infty} f(\infty; u)$. It establishes calculus rules and fundamental links to the Lipschitz-at-infinity property, connecting infinity-behavior to variational-analytic tools such as limits of normal cones and epigraphs. The authors demonstrate applications to nonsmooth optimization, deriving directional optimality conditions at infinity, coercivity, compactness of solution sets, weak sharp minima at infinity, and error bounds at infinity, aided by illustrative examples and comparisons with existing results. Overall, the paper broadens variational analysis at infinity by providing directional notions that enable precise analysis of unbounded problems and their stability properties.
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.