Solvers for Quadratically Constrained Quadratic Programs (QCQP) with complex variables

Question

I'd like to know whether there are any publicly available tools for solving QCQP with complex variables (and constraints therefore expressed through Hermitian matrices). What I have found so far is the following:

• A recently created CVXPY package for QCQP with real variables, as well as a paper mentioning the ability of BARON to handle those problems.
• Two papers that implement solvers for some QCQP problems with complex variables. Their code seems at first glance not to be available online, though.

Answer 1

Let's consider the following minimization problem

\begin{align} & \text{minimize} && \tfrac12 z^\mathrm{H} P_0 z + q_0^\mathrm{H} z\\ & \text{subject to} && \tfrac12 z^\mathrm{H} P_i z + q_i^\mathrm{H} z + r_i \leq 0 \quad \text{for } i = 1,\dots,m , \\ &&& Az = b, \end{align}

where $P_i$ are Hermitian matrices. The terms $q_i^H z$ can be complex in general, so they should not appear for us to have real-valued objective function and inequality constraints. We have then

\begin{align} & \text{minimize} && \tfrac12 z^\mathrm{H} P_0 z\\ & \text{subject to} && \tfrac12 z^\mathrm{H} P_i z + r_i \leq 0 \quad \text{for } i = 1,\dots,m , \\ &&& Az = b, \end{align} with $r_i$ real numbers.

Now, since $P_i$ is Hermitian, the following holds

$$\mathrm{Re}(P_i) = \mathrm{Re}(P_i)^T\, ,\quad \mathrm{Im}(P_i)=-\mathrm{Im}(P_i)\, ,$$

and we can write

$$z^H P_i z = \begin{pmatrix} x &y\end{pmatrix} \begin{bmatrix} \mathrm{Re}(P_i) &-\mathrm{Im}(P_i)\\ \mathrm{Im}(P_i) &\mathrm{Re}(P_i)\end{bmatrix} \begin{pmatrix} x \\y\end{pmatrix} \equiv X^T Q_i X\, , $$

where $z = x + iy$. The crossed terms cancel out due to skew-symmetry in the imaginary part and symmetry in the real part of the matrix.

Rewriting the original problem, we are left with

\begin{align} & \text{minimize} && \tfrac12 X^\mathrm{T} Q_0 X\\ & \text{subject to} && \tfrac12 X^\mathrm{T} Q_i X + r_i \leq 0 \quad \text{for } i = 1,\dots,m , \\ &&& MX = B, \end{align}

with

\begin{align} &M = \begin{bmatrix} \mathrm{Re}(A) &-\mathrm{Im}(A)\\ \mathrm{Im}(A) &\mathrm{Re}(A)\end{bmatrix}\, ,\\ &B = \begin{pmatrix} \mathrm{Re}(b) \\-\mathrm{Im}(b)\end{pmatrix}\, . \end{align}