I found a highly cited IEEE paper that optimizes over a complex optimization variable. However, they are only dealing with the magnitude of the this complex random variable multiplied by other complex constant in their problem. But, I don't understand how to show a complex problem to be convex? and how to solve it? Can I simply solve it using the lagrangian duality? I have a similar problem as follows,
$ \underset{x}{\text{Maximize}} \; \; |b^* x|$
S.T.
$ x^* x \leq 1 \;\;\;\;\;\;\;\;\;$ %that's a magnitude square constraint on x
$|a^* \; x| = 0 $
where x is a $M \times 1$ complex optimization variable, $b$ is $M \times 1$ complex constant vector, $a$ is $M \times 1$ complex constant vector as well.
the IEEE Paper with similar formulation where they highlighted that the problem is convex, problem 6 in: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4557080
Your help is very appreciated.
Thanks.