# APPENDIX

# D Method of Lagrange Multipliers

## D.1 Optimization Involving a Single Equality Constraint

Consider the minimization of a real-valued function ƒ(**w**) that is a quadratic function of a parameter vector **w**, subject to the *constraint*

where **s** is a prescribed vector and *g* is a complex constant; the superscript ^{} denotes Hermitian transposition. We may redefine the constraint by introducing a new function *c*(**w**) that is linear in **w**, as shown by

In general, the vectors **w** and **s** and the function *c*(**w**) are all *complex*. For example, in a beamforming application, the vector **w** represents a set of complex weights applied to the individual sensor outputs and **s** represents a steering vector whose elements are defined by a prescribed “look” direction; the function ƒ(**w**) to be minimized represents the mean-square value of the overall beamformer output. In a harmonic retrieval application, for another example, **w** represents the tap-weight vector of an FIR filter and **s** represents a sinusoidal vector whose elements are determined by the angular frequency of a complex sinusoid contained in the filter input; the function ƒ(**w**) represents the mean-square value of the filter output. In any event, assuming ...