In this chapter, we study specific rank-1 decomposition techniques for Hermitian positive semidefinite matrices. Based on the semidefinite programming relaxation method and the decomposition techniques, we identify several classes of quadratically constrained quadratic programming problems that are polynomially solvable. Typically, such problems do not have too many constraints. As an example, we demonstrate how to apply the new techniques to solve an optimal code design problem arising from radar signal processing.

Introduction and notation

Semidefinite programming (SDP) is a relatively new subject of research in optimization. Its success has caused major excitement in the field. One is referred to Boyd and Vandenberghe [11] for an excellent introduction to SDP and its applications. In this chapter, we shall elaborate on a special application of SDP for solving quadratically constrained quadratic programming (QCQP) problems. The techniques we shall introduce are related to how a positive semidefinite matrix can be decomposed into a sum of rank-1 positive semidefinite matrices, in a specific way that helps to solve nonconvex quadratic optimization with quadratic constraints. The advantage of the method is that the convexity of the original quadratic optimization problem becomes irrelevant; only the number of constraints is important for the method to be effective. We further present a study on how this method helps to solve a radar code design problem. Through this investigation, we aim to make a case that solving nonconvex quadratic optimization by SDP is a viable approach.