Efficient iterative solutions to complex-valued nonlinear least-squares problems with mixed linear and antilinear operators

We consider a setting in which it is desired to find an optimal complex vector x ∈ [Formula: see text] (N) that satisfies [Formula: see text] (x) ≈ b in a least-squares sense, where b ∈ [Formula: see text] (M) is a data vector (possibly noise-corrupted), and [Formula: see text] (·) : [Formula: see text] (N) → [Formula: see text] (M) is a measurement operator. If [Formula: see text] (·) 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 [Formula: see text] (·) 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 x as a vector in [Formula: see text] (2N) instead of [Formula: see text] (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 providedtodemonstratethatthisapproachcansimplifytheimplementationandreduce the computational complexity of iterative solution algorithms.