Shuzhong Zhang

Bio: Shuzhong Zhang is an academic researcher from University of Minnesota. The author has contributed to research in topics: Convex optimization & Semidefinite programming. The author has an hindex of 48, co-authored 262 publications receiving 10583 citations. Previous affiliations of Shuzhong Zhang include Fudan University & The Chinese University of Hong Kong.

Semidefinite Relaxation of Quadratic Optimization Problems

Abstract: In this article, we have provided general, comprehensive coverage of the SDR technique, from its practical deployments and scope of applicability to key theoretical results. We have also showcased several representative applications, namely MIMO detection, B? shimming in MRI, and sensor network localization. Another important application, namely downlink transmit beamforming, is described in [1]. Due to space limitations, we are unable to cover many other beautiful applications of the SDR technique, although we have done our best to illustrate the key intuitive ideas that resulted in those applications. We hope that this introductory article will serve as a good starting point for readers who would like to apply the SDR technique to their applications, and to locate specific references either in applications or theory.

Dynamic Spectrum Management: Complexity and Duality

Abstract: Consider a communication system whereby multiple users share a common frequency band and must choose their transmit power spectral densities dynamically in response to physical channel conditions. Due to co-channel interference, the achievable data rate of each user depends on not only the power spectral density of its own, but also those of others in the system. Given any channel condition and assuming Gaussian signaling, we consider the problem to jointly determine all users' power spectral densities so as to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. For the discretized version of this nonconvex problem, we characterize its computational complexity by establishing the NP-hardness under various practical settings, and identify subclasses of the problem that are solvable in polynomial time. Moreover, we consider the Lagrangian dual relaxation of this nonconvex problem. Using the Lyapunov theorem in functional analysis, we rigorously prove a result first discovered by Yu and Lui (2006) that there is a zero duality gap for the continuous (Lebesgue integral) formulation. Moreover, we show that the duality gap for the discrete formulation vanishes asymptotically as the size of discretization decreases to zero.

On cones of nonnegative quadratic functions

Abstract: We derive linear matrix inequality (LMI) characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co-positivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) level-set of a quadratic function and a half-plane. Consequently, we arrive at a generalization of Yakubovich's S-procedure result. Although the primary concern of this paper is to characterize the matrix cones by LMIs, we show, as an application of our results, that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming. Other applications are in control theory and robust optimization.

New Results on Quadratic Minimization

Abstract: In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case in which the constraints are two quadratic inequalities. This formulation is termed the extended trust region subproblem in this paper, to distinguish it from the ordinary trust region subproblem, in which the constraint is a single ellipsoid. The computational complexity of the extended trust region subproblem in general is still unknown. In this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value, and consequently we obtain polynomial-time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution. Combining this with a result obtained in the first part of the paper, we propose a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.

Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints

Abstract: We consider the NP-hard problem of finding a minimum norm vector in $n$-dimensional real or complex Euclidean space, subject to $m$ concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an $O(m^2)$ approximation in the real case and an $O(m)$ approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank $1$ (namely, outer products of some given so-called steering vectors) and the phase spread of the entries of these steering vectors are bounded away from $\pi/2$, we establish a certain “constant factor” approximation (depending on the phase spread but independent of $m$ and $n$) for both the SDP relaxation and a convex QP restriction of the original NP-hard problem. Finally, we consider a related problem of finding a maximum norm vector subject to $m$ convex homogeneous quadratic constraints. We show that an SDP relaxation for this nonconvex QP provides an $O(1/\ln(m))$ approximation, which is analogous to a result of Nemirovski e [Math. Program., 86 (1999), pp. 463-473] for the real case.

Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones

Abstract: SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.

Semidefinite Relaxation of Quadratic Optimization Problems

Abstract: In this article, we have provided general, comprehensive coverage of the SDR technique, from its practical deployments and scope of applicability to key theoretical results. We have also showcased several representative applications, namely MIMO detection, B? shimming in MRI, and sensor network localization. Another important application, namely downlink transmit beamforming, is described in [1]. Due to space limitations, we are unable to cover many other beautiful applications of the SDR technique, although we have done our best to illustrate the key intuitive ideas that resulted in those applications. We hope that this introductory article will serve as a good starting point for readers who would like to apply the SDR technique to their applications, and to locate specific references either in applications or theory.

Multi-Cell MIMO Cooperative Networks: A New Look at Interference

Abstract: This paper presents an overview of the theory and currently known techniques for multi-cell MIMO (multiple input multiple output) cooperation in wireless networks. In dense networks where interference emerges as the key capacity-limiting factor, multi-cell cooperation can dramatically improve the system performance. Remarkably, such techniques literally exploit inter-cell interference by allowing the user data to be jointly processed by several interfering base stations, thus mimicking the benefits of a large virtual MIMO array. Multi-cell MIMO cooperation concepts are examined from different perspectives, including an examination of the fundamental information-theoretic limits, a review of the coding and signal processing algorithmic developments, and, going beyond that, consideration of very practical issues related to scalability and system-level integration. A few promising and quite fundamental research avenues are also suggested.

SDPT3 — A Matlab software package for semidefinite programming, Version 1.3

Abstract: This software package is a MATLAB implementation of infeasible path-following algorithms for solving standard semidefinite programs (SDP). Mehrotra-type predictor-corrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a hundred.