  Recent developments in optimization theory have extended some traditional
algorithms for least-squares optimization of real-valued functions
(Gauss-Newton, Levenberg-Marquardt, etc.) into the domain of complex functions
of a complex variable. This employs a formalism called the Wirtinger
derivative, and derives a full-complex Jacobian counterpart to the conventional
real Jacobian. We apply these developments to the problem of radio
interferometric gain calibration, and show how the general complex Jacobian
formalism, when combined with conventional optimization approaches, yields a
whole new family of calibration algorithms, including those for the polarized
and direction-dependent gain regime. We further extend the Wirtinger calculus
to an operator-based matrix calculus for describing the polarized calibration
regime. Using approximate matrix inversion results in computationally efficient
implementations; we show that some recently proposed calibration algorithms
such as StefCal and peeling can be understood as special cases of this, and
place them in the context of the general formalism. Finally, we present an
implementation and some applied results of CohJones, another specialized
direction-dependent calibration algorithm derived from the formalism.
