Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Exact SDP relaxations for quadratic programs with bipartite graph structures

“…It is a polynomial-time method as the systems can be represented as SDPs. This result can be regarded as an extension of Azuma et al [3] in the sense that the aggregated sparsity pattern was generalized from forests to bipartite. We should mention that the signs of data are irrelavant.…”

“…In this paper, we present a wider class of QCQPs that can be solved exactly with the SDP relaxation by extending the results in [3] and [22]. The extension is based on that trees or forests are bipartite graphs and that QCQPs with no structure and the same sign of Q p ij for p = 0, 1, .…”

“…Nonconvex QCQPs (P) are known to be NP-hard in general, however, finding the exact solution of some class of QCQPs has been a popular subject [3,5,10,11,22,23,24] as they can provide solutions for important applications formulated as QCQPs (P). They include optimal power flow problems [15,28], pooling problems [14], sensor network localization problems [4,13,21], quadratic assignment problems [19,27], the max-cut problem [7].…”

“…. , Q m as in [11,22] and graph structures such as forests [3] and bipartite structures [22] have been used to identify the classes of nonconvex QCQPs whose exact optimal solution can be attained via the SDP relaxation. QCQPs with nonpositive off-diagonal data matrices were shown to have an exact SDP and SOCP relaxation [11].…”

“…• For disconnected pattern graphs, a perturbation of the objective function is introduced, as in [3], in section 4 to demonstrate that a QCQP is exact if there exists a sequence of perturbed problems converging to the QCQP while maintaining the exactness of their SDP relaxation under assumptions weaker than [3].…”