On the convex hull of convex quadratic optimization problems with indicators
This work addresses a foundational problem in mathematical optimization for researchers and practitioners, offering incremental advancements by extending and unifying existing convex hull theories.
The paper tackles the convex hull description of convex quadratic optimization problems with indicator variables, showing that the convexification reduces to a polyhedral set in an extended space with a quadratic number of variables, and provides compact mixed-integer linear formulations and descriptions in the original space. It unifies previous results and enables polyhedral methods for analyzing mixed-integer nonlinear sets.
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.