  This chapter presents reduced-rank linearly constrained minimum variance
(LCMV) algorithms based on the concept of joint iterative optimization of
parameters. The proposed reduced-rank scheme is based on a constrained robust
joint iterative optimization (RJIO) of parameters according to the minimum
variance criterion. The robust optimization procedure adjusts the parameters of
a rank-reduction matrix, a reduced-rank beamformer and the diagonal loading in
an alternating manner. LCMV expressions are developed for the design of the
rank-reduction matrix and the reduced-rank beamformer. Stochastic gradient and
recursive least-squares adaptive algorithms are then devised for an efficient
implementation of the RJIO robust beamforming technique. Simulations for a
application in the presence of uncertainties show that the RJIO scheme and
algorithms outperform in convergence and tracking performances existing
algorithms while requiring a comparable complexity.
