On Valid Inequalities for Quadratic Programming with Continuous Variables and Binary Indicators

Abstract

Abstract. In this paper we study valid inequalities for a set that involves a continuous vector variable x ∈ [0, 1] n , its associated quadratic form xx T , and binary indicators on whether or not x > 0. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). Valid inequalities for this set can be obtained by lifting inequalities for a related set without binary variables (QPB), that was studied by Burer and Letchford. After closing a theoretical gap about QPB, we characterize the strength of different classes of lifted QPB inequalities. We show that one class, lifted-posdiag-QPB inequalities, capture no new information from the binary indicators. However, we demonstrate the importance of the other class, called lifted-concave-QPB inequalities, in two ways. First, all lifted-concave-QPB inequalities define the relevant convex hull for the case of convex quadratic programming with indicators. Second, we show that all perspective constraints are a special case of lifted-concave-QPB inequalities, and we further show that adding the perspective constraints to a semidefinite programming (SDP) relaxation of convex quadratic programs with binary indicators results in a problem whose bound is equivalent to the recent optimal diagonal splitting approach of Zheng et al.. Finally, we show the separation problem for lifted-concave-QPB inequalities is tractable if the number of binary variables involved in the inequality is small. Our study points out a direction to generalize perspective cuts to deal with non-separable nonconvex quadratic functions with indicators in global optimization. Several interesting questions arise from our results, which we detail in our concluding section.

Strong formulations for quadratic optimization with M-matrices and indicator variables

“…We now show how to project out the additional variables (x,ȳ,t), (x,ŷ,t) to find conv(X 1 ) in the original space of variables, which can be done directly from the non-convex formulation above. From constraints (14) and (17) we see λ = x, from constraint (15)ŷ 1 = y 1 x , from (18) y 1 ≤ x, from (16) we findȳ 2 = y 2 −xŷ 2 1−x , and from (19) we get 0 ≤ŷ 2 ≤ 1 and 0 ≤ y 2 −xŷ 2 1−x ≤ 1. Thus, (13)- (21) is feasible if and only if 0 ≤ y 1 ≤ x, 0 ≤ y 2 ≤ 1 and there existsŷ 2 such that…”

“…There are numerous approaches in the literature for deriving strong formulations for (QOI) and S. Dong and Linderoth [19] describe lifted inequalities for (QOI) from its continuous quadratic optimization counterpart over bounded variables. Bienstock and Michalka [12] give a characterization linear inequalities obtained by strengthening gradient inequalities of a convex objective function over a nonconvex set.…”

Relaxing Nonconvex Quadratic Functions by Multiple Adaptive Diagonal Perturbations

“…[RRW10] provides a review of solution approaches for this problem and proposes to incorporate some strong SDP-based relaxations by solving them using bundle method. Another relevant line of research [FG07,GL10,ZSL10,DL13] focused on globally solving convex quadratic programming with binary indicator variables, when combined with the so-called perspective constraints, diagonal perturbations are also shown to be very important.…”

Extended formulations in mixed integer conic quadratic programming

“…If the nonlinear constraints are not satisfied, the algorithm first attempts to find a feasible solution to MICQP that has the same values in the integer variables as the solution to LP l, u, Γ l k l=1 for the current node. This is done by solving CP (l, u) for appropriately chosen bounds in lines [15][16][17]. If this heuristic is successful and yields a better solution the incumbent is updated in lines 18-20.…”

“…that does not use L d ε or L d s(ε) ), we simply solve the three lifted reformulations described in Section 4.1 with CPLEX and Gurobi's LP-based algorithms. We refer to the implementations based on reformulation (15) as CPLEXTowerLP and GurobiTowerLP, to the implementations based on reformulation (16) as CPLEXSe-pLP and GurobiSepLP and to the implementations based on reformulation (17) as CPLEXTowerSepLP and GurobiTowerSepLP.…”