  A method is proposed for solving equality constrained nonlinear optimization
problems involving twice continuously differentiable functions. The method
employs a trust funnel approach consisting of two phases: a first phase to
locate an $\epsilon$-feasible point and a second phase to seek optimality while
maintaining at least $\epsilon$-feasibility. A two-phase approach of this kind
based on a cubic regularization methodology was recently proposed along with a
supporting worst-case iteration complexity analysis. Unfortunately, however, in
that approach, the objective function is completely ignored in the first phase
when $\epsilon$-feasibility is sought. The main contribution of the method
proposed in this paper is that the same worst-case iteration complexity is
achieved, but with a first phase that also accounts for improvements in the
objective function. As such, the method typically requires fewer iterations in
the second phase, as the results of numerical experiments demonstrate.
