# Optimization in complex spaces with the Mixed Newton Method

arXiv: https://arxiv.org/abs/2207.04265 · HTML: https://mlsanity.com/paper/2207.04265/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2022.

## Built on by (most cited first, 1 of 1)

- [Mixed Newton Method for Optimization in Complex Spaces](https://mlsanity.com/paper/2407.20367/) (2024)

## Abstract (arXiv)

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 fracpartial2fpartialbar zpartial z for computation of the search direction, as opposed to the full Hessian fracpartial2fpartial(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.