Supermodularity and valid inequalities for quadratic optimization with indicators
研究了带指示变量的秩一二次函数最小化问题，发现投影连续变量后的集合函数是超模的，并给出了凸包描述的显式形式及多项式分离算法，实验表明所提不等式能有效缩小整数间隙。
Abstract 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 obtained 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.