Exploring constraint qualification-free optimality conditions for linear second-order cone programming

Abstract

Linear second-order cone programming (SOCP) deals with optimization problems characterized by a linear objective function and a feasible region defined by linear equalities and second-order cone constraints. These constraints involve the norm of a linear combination of variables, enabling the representation of a wide range of convex sets. The SOCP serves as a potent tool for addressing optimization challenges across engineering, finance, machine learning, and various other domains. In this paper, we introduce new optimality conditions tailored for {SOCP} problems. These conditions have the form of two optimality criteria that are obtained without the requirement of any constraint qualifications and are defined explicitly. The first criterion utilizes the concept of immobile indices of constraints. The second criterion, without relying explicitly on immobile indices, introduces a special finite vector set for assessing optimality. To demonstrate the effectiveness of these criteria, we present two illustrative examples highlighting their applicability. We compare the obtained criteria with other known optimality conditions and show the advantage of the former ones.