Lower bounds for a constrained quadratic 0-1 problem

Cheng Zhu Lu, Zhenbo Wang, Wenxun Xing

Zi Li Xu, Mingyi Hong

Guangtai Qu, Shaoze Li, Zhibin Deng +1 more

The multi-linearly constrained separable binary quadratic programming problem (MLSBQP) is a direct extension of the classical singly-linearly constrained separable binary quadratic programming problem (SLSBQP). In the literature, several algorithms have been proposed for SLSBQP which can solve problem instances with millions of variables globally. However, for the case of MLSBQP, the current state-of-the-art solver can only solve problems with up to 40000 binary variables within a reasonable computational time. In this paper, we propose an extremely efficient branch-and-bound algorithm for solving MLSBQP. The main feature of the proposed algorithm is that an efficient dual coordinate ascent algorithm is applied to obtain lower bounds, and the complexity of each inner loop of the dual coordinate ascent algorithm is $ \mathcal{O}(n\log n) $. Our numerical experiments show that the proposed branch-and-bound algorithm can solve MLSBQP with up to millions of variables within only several minutes, and performs much faster than Gurobi when the number of linear constraints is small.

Yurii E. Nesterov

Francisco Barahona, Michael Jünger, Gerhard Reinelt

Junyu Zhang, Guanyi Wang

Continuous submodular optimization is an important class of nonconvex problems with global guarantees, for which the submodular box-constrained quadratic programming (BCQP) forms a fundamental subclass. It arises naturally as quadratic subproblems of general twice-differentiable submodular problems and also has direct business and management applications such as pricing. This paper studies the worst-case relaxation gap of semidefinite programming (SDP) relaxations with a finite family of valid linear cuts for submodular BCQP. We show that in dimension $n\geq4$, for any SDP relaxation formulated with a finite family of linear cuts, there always exists a submodular BCQP instance with a strictly positive relaxation gap. By restricting to the normalized BCQP instances that remove the problem's scaling factor, we further quantify this limitation with explicit lower bounds. In general dimension $n\geq4$, when the SDP relaxation is formulated with only Boolean-quadric-polytope (BQP) valid cuts, including the commonly used McCormick and triangle inequalities, we establish an $Ω(n)$ relaxation gap lower bound. For an arbitrary family of $m$ valid linear cuts, we establish an $Ω(n/m^2)$ lower bound. These results capture the fundamental limitations of SDP relaxations, especially SDP with BQP valid cuts, and provide guidance on what additional constraints may be needed for more effective convex approximations of submodular BCQP.