This paper is dedicated to recent developments of a two-Lagrange multipliers domain decomposition method for the Helmholtz equation [C. Farhat et al., Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 213--240; M. J. Gander, F. Magoulès, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60] involving an additional augmented operator along the interface between the subdomains. Most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator [B. Després, Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman et al., eds., SIAM, Philadelphia, 1993, pp. 197--206; J.-D. Benamou and B. Després, J. Comput. Phys., 136 (1997), pp. 68--82; P. Chevalier and F. Nataf, Domain Decomposition Methods 10, AMS, Providence, RI, 1998, pp. 400--407; M. J. Gander, Proceedings of the 12th International Conference on Domain Decomposition Methods, (Chiba, Japan), ddm.org, 2000, pp. 247--254; M. J. Gander, F. Magoulès, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60]. At the discrete level, the optimal operator can be proved to be equal to the Schur complement of the outer domain. This Schur complement can be directly approximated using purely algebraic techniques like sparse approximate inverse methods or incomplete factorization. The main advantage of such an algebraic approach is that it is much easier to implement in existing code without any information on the geometry of the interface and the finite element formulation used. Convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented.