  We consider a setting in which it is desired to find an optimal complex
vector $\mathbf{x}\in\mathbb{C}^N$ that satisfies $\mathcal{A}(\mathbf{x})
\approx \mathbf{b}$ in a least-squares sense, where $\mathbf{b} \in
\mathbb{C}^M$ is a data vector (possibly noise-corrupted), and
$\mathcal{A}(\cdot): \mathbb{C}^N \rightarrow \mathbb{C}^M$ is a measurement
operator. If $\mathcal{A}(\cdot)$ were linear, this reduces to the classical
linear least-squares problem, which has a well-known analytic solution as well
as powerful iterative solution algorithms. However, instead of linear
least-squares, this work considers the more complicated scenario where
$\mathcal{A}(\cdot)$ is nonlinear, but can be represented as the summation
and/or composition of some operators that are linear and some operators that
are antilinear. Some common nonlinear operations that have this structure
include complex conjugation or taking the real-part or imaginary-part of a
complex vector. Previous literature has shown that this kind of mixed
linear/antilinear least-squares problem can be mapped into a linear
least-squares problem by considering $\mathbf{x}$ as a vector in
$\mathbb{R}^{2N}$ instead of $\mathbb{C}^N$. While this approach is valid, the
replacement of the original complex-valued optimization problem with a
real-valued optimization problem can be complicated to implement, and can also
be associated with increased computational complexity. In this work, we
describe theory and computational methods that enable mixed linear/antilinear
least-squares problems to be solved iteratively using standard linear
least-squares tools, while retaining all of the complex-valued structure of the
original inverse problem. An illustration is provided to demonstrate that this
approach can simplify the implementation and reduce the computational
complexity of iterative solution algorithms.
