Supermodularity and valid inequalities for quadratic optimization with indicators
This work provides a method for tightening the relaxation of quadratic optimization problems with indicator variables, which is important for practitioners solving such problems.
This paper addresses the minimization of rank-one quadratic functions with indicator variables, demonstrating that the projected set function is supermodular and can be minimized in linear time. The authors derive the convex hull of the epigraph of the quadratic using lifted supermodular inequalities, providing both original-space and extended conic quadratic formulations, along with a polynomial separation algorithm. Computational experiments show that the lifted supermodular inequalities significantly reduce the integrality gap for quadratic optimization with indicators.
We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtaining from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.