This paper introduces a novel family of constrained adaptive filtering algorithms for sensor array beamforming. These algorithms, namely, constrained least mean logarithmic square (CLMLS), ℓ1-norm CLMLS (ℓ1-CLMLS) and its weighted version ℓ1-WCLMLS, are developed based on a relative logarithmic cost function. The proposed algorithms gracefully adjust cost function depending on the amount of the error thereby achieving better performance compared to constrained least mean square (CLMS) family of algorithms. The transient and steady-state performance analysis of the proposed CLMLS algorithm is presented and these analytical results are validated through extensive simulations. Proposed CLMLS algorithm is then extended to sparse system identification problem by incorporating the ℓ1-norm penalty into CLMLS cost function. We show that the resultant ℓ1-CLMLS and ℓ1-WCLMLS algorithms outperform their CLMS counterparts in sparse system identification. When applied to sparse sensor array synthesis, these algorithms achieve desired beampattern with lesser number of sensor elements compared to state-of-the-art algorithms. © 2001-2012 IEEE.