The semidefinite matrix rank minimization, which has a broad range of applications in system control, statistics, network localization, econometrics and so on, is computationally NPhard in general due to the noncontinuous and non-convex rank function. A natural way to handle this type of problems is to substitute the rank function into some tractable surrogates, most popular ones of which inclu...

We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorith...

In this paper, we propose a new robust analysis tool motivated by large-scale systems. The H∞ norm of a system measures its robustness by quantifying the worst-case behavior of a system perturbed by a unit-energy disturbance. However, the disturbance that induces such worst-case behavior requires perfect coordination among all disturbance channels. Given that many systems of interest, such as t...

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension n where n is the number of nodes. To achieve a speed up, we propose a ...

We observe that in a simple one-dimensional polynomial optimization problem (POP), the ‘optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the st...

We determine the information-theoretic cutoff value on separation of cluster centers for exact recovery labels in a K-component Gaussian mixture model with equal sizes. Moreover, we show that semidefinite programming (SDP) relaxation K-means clustering method achieves such sharp threshold without assuming symmetry centers.

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x∗Cx | x∗Akx ≥ 1, x ∈ F, k = 0, 1, ...,m}; and (2) max{x∗Cx | x∗Akx ≤ 1, x ∈ F, k = 0, 1, ...,m}. If one of Ak’s is indefinite while others and C are positive semidefini...

We consider a random Integer Least Squares (ILS) problem. This NP-hard problem naturally arises in digital communications as the maximum-likelihood detection problem. We analyze two probabilistic quasi-ILS algorithms based on semidefinite relaxations: the SDR algorithm for binary variables and the PSK algorithm for constant modulus variables. Both algorithms are capable of delivering a near-opt...