Tight semidefinite programming relaxations for sparse box-constrained quadratic programs
Aida Khajavirad
TL;DR
This work develops tight SDP relaxations for sparse box-constrained quadratic programs by integrating the Reformulation Linearization Technique (RLT) into SDP relaxations and exploiting problem sparsity. It provides constructive SDP representations for $\text{QP}(G)$ under a sufficient graph-structure condition (no connected-plus-triplets) and shows how to assemble extended formulations from component subproblems, building on prior work on SOC/SDP relaxations. A key contribution is identifying conditions under which these relaxations are SDP-representable with polynomial size and polynomial-time construction, notably when the graph has bounded treewidth and bounded degree on plus-loop nodes. The results generalize existing SOC relaxations and offer scalable SDP frameworks for sparse nonconvex QPs, with precise trade-offs between relaxation strength and computational size.
Abstract
We introduce a new class of semidefinite programming (SDP) relaxations for sparse box-constrained quadratic programs, obtained by a novel integration of the Reformulation Linearization Technique into standard SDP relaxations while explicitly exploiting the sparsity of the problem. The resulting relaxations are not implied by the existing LP and SDP relaxations for this class of optimization problems. We establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SDP-representable; the proof is constructive and yields an explicit extended formulation. Although the resulting SDP may be of exponential size in general, we further identify additional structural conditions on the sparsity of the optimization problem that guarantee the existence of a polynomial-size SDP-representable formulation, which can be constructed in polynomial time.