  A well-known challenge in beamforming is how to optimally utilize the degrees
of freedom (DoF) of the array to design a robust beamformer, especially when
the array DoF is limited. In this paper, we leverage the tool of constrained
convex optimization and propose a penalized inequality-constrained minimum
variance (P-ICMV) beamformer to address this challenge. Specifically, a
well-targeted objective function and inequality constraints are proposed to
achieve the design goals. By penalizing the maximum gain of the beamformer at
any interfering directions, the total interference power can be efficiently
mitigated with limited DoF. Multiple robust constraints on the target
protection and interference suppression can be introduced to increase the
robustness of the beamformer against steering vector mismatch. By integrating
the noise reduction, interference suppression, and target protection, the
proposed formulation can efficiently obtain a robust beamformer design while
optimally trading off various design goals. To numerically solve this problem,
we formulate the P-ICMV beamformer design as a convex second-order cone program
(SOCP) and propose a low complexity iterative algorithm based on the
alternating direction method of multipliers (ADMM). Three applications are
simulated to demonstrate the effectiveness of the proposed beamformer.
