This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers' failure.
H. L. Hijazi and L.Liberti "Constraint Qualification Failure in Action" Operations Research Letters (2016) 44(1) : 503 − 506