In this paper, we present a new discrete Lagrangian optimization method for designing multiplierless QMF (quadrature mirror filter) filter banks. In multiplierless QMF filter banks, filter coefficients are powers-of-two (PO2) where numbers are represented as sums or differences of powers of two (also called Canonical Signed Digit-CSD-representation), and multiplications can be carried out as additions, subtractions and shifting. We formulate the design problem as a nonlinear discrete constrained optimization problem, using the reconstruction error as the objective, and other performance metrics as constraints. One of the major advantages of this formulation is that it allows us to search for designs that improve over the best existing designs with respect to all performance metrics, rather than finding designs that trade one performance metric for another. We show that our design method can find designs that improve over Johnston's benchmark designs using a maximum of three to six ONE bits in each filter coefficient.