TY - JOUR

T1 - Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints

AU - Thng, I.

AU - Cantoni, A.

AU - Leung, Y. H.

PY - 1996/9/1

Y1 - 1996/9/1

N2 - In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50-90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional 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 decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate 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 manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.

AB - In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50-90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional 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 decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate 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 manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.

UR - http://www.scopus.com/inward/record.url?scp=0030241002&partnerID=8YFLogxK

U2 - 10.1007/s002459900025

DO - 10.1007/s002459900025

M3 - Article

AN - SCOPUS:0030241002

SN - 0095-4616

VL - 34

SP - 161

EP - 182

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

IS - 2

ER -