ANALYTICAL SOLUTIONS TO THE OPTIMIZATION OF A QUADRATIC COST FUNCTIONSUBJECT TO LINEAR AND QUADRATIC EQUALITY CONSTRAINTS

In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimizatio n problem is often required. The problem posed is usually characterize d by a large optimization space (around 50-90 tuples), a large number of linear equality constraints, and a few quadratic equality constrain ts each having very low rank quadratic constraint matrices. Two main d ifficulties arise in this class of problem. Firstly, the feasibility r egion is nonconvex and multiple local minima abound. This makes conven tional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of d ecision-method algorithms for the theory of the reals unattractive. Th is is because these algorithms involve the solution of the roots of un ivariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more managea ble size. The new algorithm relies on linear-algebra concepts, basic o ptimization theory, and a multivariate polynomial root-solving tool of ten used by decision-method algorithms.