  The full-waveform inversion (FWI) addresses the computation and
characterization of subsurface model parameters by matching predicted data to
observed seismograms in the frame of nonlinear optimization. We formulate FWI
as a nonlinearly constrained optimization problem, for which a regularization
term is minimized subject to the nonlinear data matching constraint. Unlike FWI
which is based on the penalty function, the method of multipliers solves the
resulting optimization problems by using the augmented Lagrangian function; and
leads to a two-step recursive algorithm. The primal step requires solving an
unconstrained minimization problem like the traditional FWI with a difference
that the data are replaced by the Lagrange multipliers. The dual step involves
an update of the Lagrange multipliers. The overall performance of the algorithm
is improved considering that this multiplier method does not require an exact
solution of these primal-dual subproblems. In fact, convergence is attained
when only one step of a gradient-based method is taken on both subproblems. The
proposed algorithm greatly improves the overall performance of FWI such as
convergence from inaccurate starting models and robustness with respect to the
determination of the step length. Furthermore, it can be performed by the
existing FWI engines with minimal change. We only have to replace the observed
data at each iteration with the multipliers, thus all the nice properties of
the traditional FWI algorithms are kept. Numerical experiments confirm that the
multipliers waveform inversion can converge to a solution of the inverse
problem in the absence of low-frequency data from an inaccurate initial model
even with a constant step size.
