Shuzhong Zhang

Semidefinite Relaxation of Quadratic Optimization Problems

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.

On cones of nonnegative quadratic functions

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 Yakubovichs 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

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

We consider the NP-hard problem of finding a minimum norm vector in

Complex Matrix Decomposition and Quadratic Programming

n

Quadratic maximization and semidefinite relaxation

-dimensional real or complex Euclidean space, subject to

High Performance Optimization

m

New results on Hermitian matrix rank-one decomposition

concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an

Complex Quadratic Optimization and Semidefinite Programming

O(m^2)

Pivot rules for linear programming: A survey on recent theoretical developments

approximation in the real case and an