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Binary quadratic optimization problems that are difficult to solve by conic relaxations. (English) Zbl 1387.90175
Summary: We study conic relaxations including semidefinite programming (SDP) relaxations and doubly nonnegative programming (DNN) relaxations to find the optimal values of binary QOPs. The main focus of the study is on how the relaxations perform with respect to the rank of the coefficient matrix in the objective of a binary QOP. More precisely, for a class of binary QOP instances, which include the max-cut problem of a graph with an odd number of nodes and equal weight, we show numerically that (1) neither the standard DNN relaxation nor the DNN relaxation derived from the completely positive formulation by Burer performs better than the standard SDP relaxation, and (2) Lasserre’s hierarchy of SDP relaxations requires solving the SDP with the relaxation order at least \(\lceil n/2\rceil\) to attain the optimal value. The bound \(\lceil n/2\rceil\) for the max-cut problem of a graph with equal weight is consistent with M. Laurent’s conjecture in [Math. Oper. Res. 28, No. 4, 871–883 (2003; Zbl 1082.90085)], which was proved recently by H. Fawzi et al. in [“Sparse sum-of-squares certificates on finite abelian groups”, Preprint, arXiv:1503.01207].

MSC:
90C20 Quadratic programming
90C22 Semidefinite programming
90C10 Integer programming
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