Consider the least squares optimization problem:
$$ \min_{ \mathbf{x} \in \mathbb{C}^M} \left\lVert \mathbf{y} - \mathbf{Ax}\right\rVert_2. $$
I want to impose constraints on the relative phase of successive elements of $\mathbf{x}$. Obviously, if want to fix the relative phase, I can consider constraints of the form:
$$x_i = e^{j\theta_i}x_{i+1}, \quad i=1,2,...,M-1,$$
which is a fairly simple linear constraint (this also constrains the magnitude though). Instead of this, however, I would like to constrain the relative phase to fall into some range.
Is it possible to represent this constraint by a series of simple convex constraints?
Here is my attempt:
Say we want to constrain the relative phase of $x_{i+1}$ and $x_i$such that it falls between $\theta_{min}$ and $\theta_{max}$ (also assume $\theta_{max} - \theta_{min} \leq \pi$). We write $x_{i} = a_{i}+ jb_{i}$. This way, a phase constraint in $\mathbb{C}^1$ is essentially in cone constraint in $\mathbb{R}^2$. Any cone in $\mathbb{R}^2$ can be written as $\mathbf{Ms \geq 0}$ where $\geq$ is defined as a generalized inequality based on the non-negative quadrant (i.e each element should be non-zero). In this case, this matrix is (I think): $$ \mathbf{M}= \begin{bmatrix} \cos(\theta_{min} + \frac{\pi}{2} )a_i - \sin(\theta_{min} + \frac{\pi}{2})b_i & \sin(\theta_{min} + \frac{\pi}{2} )a_i + \cos(\theta_{min} + \frac{\pi}{2})b_i \\ \cos(\theta_{max} - \frac{\pi}{2} )a_i - \sin(\theta_{max} - \frac{\pi}{2})b_i & \sin(\theta_{max} - \frac{\pi}{2} )a_i + \cos(\theta_{max} - \frac{\pi}{2})b_i \\ \end{bmatrix}. $$
Simplifying, we have $$ M = \begin{bmatrix} \begin{bmatrix} a_i & b_i \end{bmatrix} R(\theta_{min}+ \frac{\pi}{2}) \\ \begin{bmatrix} a_i & b_i \end{bmatrix} R(\theta_{max} - \frac{\pi}{2}) \\ \end{bmatrix} $$ where R is the standard rotation matrix. So our overall constraints are: $$\begin{bmatrix} a_i & b_i \end{bmatrix} R\left(\theta_{min}+ \frac{\pi}{2}\right) \begin{bmatrix} a_{i+1} \\ b_{i+1} \end{bmatrix} \geq 0 $$
$$ \begin{bmatrix} a_i & b_i \end{bmatrix} R\left(\theta_{max} - \frac{\pi}{2}\right) \begin{bmatrix} a_{i+1} \\ b_{i+1} \end{bmatrix} \geq 0 $$ These don't look convex, though I feel as though it might yet be possible to write these constraints in some convex way. It is clear that these cones are convex, but the cast that we are considering a relative phase may be an issue. Any help would be greatly appreciated.