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 … Read more
We study the convex hull of the mixed-integer set given by a conic quadratic inequality and indicator variables. Conic quadratic terms are often used to encode uncertainties, while the indicator variables are used to model fixed costs or enforce sparsity in the solutions. We provide the convex hull description of the set under consideration when … Read more
We study quadratic optimization with semi-continuous variables and an M-matrix, i.e., PSD with non-positive off-diagonal entries. This structure arises in image segmentation, portfolio optimization, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular … Read more