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In this paper, we study the mixed-integer nonlinear set given by a separablequadratic constraint on continuous variables, where each continuous variable iscontrolled by an additional indicator. This set occurs pervasively inoptimization problems with uncertainty and in machine learning. We show thatoptimization over this set is NP-hard. Despite this negative result, wecharacterize the structure of the convex hull, and show that it can be formallystudied using polyhedral theory. Moreover, we show that although perspectiverelaxation in the literature for this set fails to match the structure of itsconvex hull, it is guaranteed to be a close approximation.

A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation