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<p>In this paper, we first establish some properties of normal cones with respect to a set and derive calculus rules for subdifferentials with respect to a set at a reference point. Next, we introduce the concepts of normal cone with respect to a set to an epigraph at infinity, and subdifferential with respect to a set at infinity. Then, several properties of these notions at infinity are explored. Finally, we examine these tools at infinity to obtain necessary optimality conditions at infinity, the compactness of the optimal solution set, and the coercivity of the objective function for the underlying optimization problem under the unboundedness of its associated feasible set.</p>

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Normal Cone and Subdifferential with Respect to a Set at Infinity and Their Applications

  • 期刊论文

  • 发表日期：

    2025年6月27日

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  • Nguyen Le Hoang Anh,
  • Nguyen Canh Hung

摘要

In this paper, we first establish some properties of normal cones with respect to a set and derive calculus rules for subdifferentials with respect to a set at a reference point. Next, we introduce the concepts of normal cone with respect to a set to an epigraph at infinity, and subdifferential with respect to a set at infinity. Then, several properties of these notions at infinity are explored. Finally, we examine these tools at infinity to obtain necessary optimality conditions at infinity, the compactness of the optimal solution set, and the coercivity of the objective function for the underlying optimization problem under the unboundedness of its associated feasible set.