What is the solution for the following minimization problem?

Question

Given vectors $\mathbb{y}, \mathbb{b} \in \mathbb C^n$, find

$$\arg\min_{x \in \mathbb R} ~ \| \mathbb{y} - \exp(-jx) \mathbb{b} \|^2$$

Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

Rewriting as a complex QCQP in $z \in \mathbb C$,

$$\begin{array}{ll} \text{minimize} & \| \mathrm a \, z - \mathrm b \|_2^2\\ \text{subject to} & z^* z = 1\end{array}$$

where $\mathrm a, \mathrm b \in \mathbb C^n$ are given. Note that

$$\| \mathrm a \, z - \mathrm b \|_2^2 = \left( \mathrm a \, z - \mathrm b \right)^* \left( \mathrm a \, z - \mathrm b \right) = \| \mathrm a \|_2^2 \, \underbrace{|z|^2}_{=1} + \| \mathrm b \|_2^2 - \left( \mathrm b^* \mathrm a \, z + \mathrm a^* \mathrm b \, z^* \right)$$

Let $c := \mathrm a^* \mathrm b =: \langle \mathrm a, \mathrm b\rangle$. Hence,

$$\| \mathrm a \, z - \mathrm b \|_2^2 = \| \mathrm a \|_2^2 + \| \mathrm b \|_2^2 - \left( c^* z + c \, z^* \right) = \| \mathrm a \|_2^2 + \| \mathrm b \|_2^2 - 2 \Re \{ c^* z \}$$

which is minimal when $\Re \{ c^* z \}$ is maximal. Let $\theta := \arg(c)$. Thus, the optimum is at $z = e^{i\theta}$.