Quadratic Optimisation with One Quadratic Equality Constraint

U) This report presents a theoretical framework for minimising a quadratic objective function subject to a quadratic equality constraint. The first part of the report gives a detailed algorithm which computes the global minimiser without calling special nonlinear optimisation solvers. The second part of the report shows how the developed theory can be applied to solve the time of arrival geolocation problem. RELEASE LIMITATION Approved for public release Published by Electronic Warfare and Radar Division DSTO Defence Science and Technology Organisation PO Box 1500 Edinburgh South Australia 5111 Australia Telephone: (08)7389 5555 Fax: (08) 7389 6567 © Commonwealth of Australia 2010 AR-014-772 June 2010 APPROVED FOR PUBLIC RELEASE Quadratic Optimisation with One Quadratic Equality Constraint Executive Summary The theoretical work presented in this report is motivated by the need to solve many defence application problems such as the time of arrival geolocation problem. The mathematical tools needed to solve such a localisation problem, are developed in detail in the first part of the report (Part I) and are based on quadratic optimization with one quadratic equality constraint. The main contribution of this work is the development of an algorithm which provides a step-by-step procedure to solve the problem and address solution feasibility and uniqueness issues. This algorithm relies heavily on linear algebraic transformations and optimality condition properties to efficiently and exactly determine the problem minimiser. The localisation of an emitter source or a receiver based on time of arrival (TOA) measurements is a demonstration example given in the second part of the report (Part II), which illustrates how the developed theory is used.