Supermodularity and valid inequalities for quadratic optimization with indicators

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 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 quadraticrepresentable 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.

A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

“…Definition 2 (SP graph). Define the weighted directed acyclic graph G SP with vertex set N ∪ {0, n + 1}, arc set A = {(i, j) ∈ Z 2 + : 0 ≤ i < j ≤ n + 1} and weights w given in (6). Figure 2 depicts a graphical representation of G SP .…”

“…Algorithm 1 is based on the forward elimination of variables. Consider the optimization problem (6), which we repeat for convenience…”

“…This convex hull characterization has led to the development of several techniques for problem (1) with general Q, including cutting plane methods [26,25], strong MISOCP formuations [1,32], approximation algorithms [56], specialized branching methods [34], and presolving methods [5]. Recently, problem (1) has been studied under other structural assumptions, including: quadratic terms involving two variables only [2,3,27,33,37], rank-one quadratic terms [4,6,51,50], and quadratic terms with Stieltjes matrices [7]. If the matrix can be factorized as Q = Q 0 Q 0 where Q 0 is sparse (but Q is dense), then problem (1) can be solved (under appropriate conditions) in polynomial time [21].…”

On the convex hull of convex quadratic optimization problems with indicators

“…Similar relaxations based on separable quadratic terms were considered in [17,33]. A generalization of the above approach is to let Γ i = h i h ⊤ i be a rank-one matrix [6,7,23,31,32]; in this case, letting…”

The equivalence of optimal perspective formulation and Shor's SDP for quadratic programs with indicator variables

“…Sufficient conditions for exactness of SDP relaxations [e.g. 9, 23,26,30,29] and stronger rank-one conic formulations [4,5] are also given in the literature.…”