  We propose a second-order method for unconditional minimization of functions
$f(z)$ of complex arguments. We call it the Mixed Newton Method due to the use
of the mixed Wirtinger derivative $\frac{\partial^2f}{\partial\bar z\partial
z}$ for computation of the search direction, as opposed to the full Hessian
$\frac{\partial^2f}{\partial(z,\bar z)^2}$ in the classical Newton method. The
method has been developed for specific applications in wireless network
communications, but its global convergence properties are shown to be superior
on a more general class of functions $f$, namely sums of squares of absolute
values of holomorphic functions. In particular, for such objective functions
minima are surrounded by attraction basins, while the iterates are repelled
from other types of critical points. We provide formulas for the asymptotic
convergence rate and show that in the scalar case the method reduces to the
well-known complex Newton method for the search of zeros of holomorphic
functions. In this case, it exhibits generically fractal global convergence
patterns.
