On the convex hull of convex quadratic optimization problems with indicators

Abstract We consider the convex quadratic optimization problem in $$\mathbb {R}^{n}$$ R n with indicator variables and arbitrary constraints on indicators. show that a hull description of associated mixed-integer set an extended space number additional consists $$(n+1) \times (n+1)$$ ( + 1 ) × positive semidefinite constraint (explicitly stated) linear constraints. In particular, convexification this class problems reduces to describing polyhedral formulation. While vertex representation is exponential explicit inequality may not be readily available general, we derive compact formulation whose solutions coincide vertices set. also give descriptions original variables: provide based infinite conic-quadratic inequalities, which are “finitely generated.” it possible characterize whether given necessary describe hull. The new theory presented here unifies several previously established results, paves way toward utilizing methods analyze nonlinear sets.