Consider the following nonlinear minimization problem \begin{align} \tag{P1} \min_{x_1, x_2 \in \mathbb C} \sum_{i=1}^m \big|a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|x_2|^2 - b_i\big|^2 \end{align} where $a_{i1}, a_{i2}, a_{i3}, a_{i4}, b_i$ are nonzero constants in $\mathbb C$, $i=1,\ldots,m$.

The first thing that occurred to me was to reformulate it with a change of variables \begin{align} \tag{P2} \min_{y_1, y_2, y_3, y_4 \in \mathbb C} & \sum_{i=1}^m \left|a_{i1}y_1 + a_{i2}y_2 + a_{i3}y_3 + a_{i4}y_4 - b_i\right|^2 \\ \text{subject to} \quad & y_1, y_4 \in \mathbb R_{\geq 0} \\ & y_2 = \overline{y_3} \\ & y_1y_4 = y_2y_3 \end{align} so that the objective function can be written more compactly in matrix form (e.g., $\|Ax-b\|_2^2$). The equivalence can be established by the result kindly proved by Batominovski here.

My questions are for example:

  • Is there any algorithm which can effectively solve (P1)?
    • If there is, is it guaranteed to converge to a global or a local minimum?
  • Can (P2) be recast as a convex problem?
    • If it cannot, is there any similar (convex) problem which can be considered?

The following attempt concerns the special case $m=1$. In this case (P2) becomes \begin{align} \tag{P3} \min_{y_1, y_2, y_3, y_4 \in \mathbb C} & \left|a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4 - b\right|^2 \\ \text{subject to} \quad & y_1, y_4 \in \mathbb R_{\geq 0} \\ & y_2 = \overline{y_3} \\ & y_1y_4 = y_2y_3 \end{align} where $a_1, a_2, a_3, a_4, b$ are nonzero constants in $\mathbb C$.

The idea is to linearize the bilinear constraint by taking the logarithm.

Assume $y_j \neq 0$, let $z_j := \ln y_j + \ln a_j$, $j=1,\ldots,4$. In order to take advantage of the convexity of the log-sum-exp function, consider a similar problem \begin{align} \tag{P4} \min_{z_1, z_2, z_3, z_4 \in \mathbb C} & \left|\ln(e^{z_1}+e^{z_2}+e^{z_3}+e^{z_4}) - \ln b\right|^2 \\ \text{subject to} \quad & z_1 - \ln a_1 \in \mathbb R, y_4 - \ln a_4 \in \mathbb R \\ & z_2 - \overline{z_3} = \ln a_2 - \ln \overline{a_3} \\ & z_1 - z_2 - z_3 + z_4 = \ln a_1 - \ln a_2 - \ln a_3 + \ln a_4 \end{align} The rationale is to take the logarithm of both sides of $b \approx a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4$. Obviously, this trick cannot be applied to the general case due to the varying $a$'s. Despite the extra assumption, I am also uncomfortable about $\ln$ being multivalued in $\mathbb C$.


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