SOCP with complex variables

I am solving solving an SOCP of the form,

minimize h’*x
subject to
h’x >= ||Hx||

where, h and H are give complex vector vector and matrix; and x is my optimization vector which is also complex. CVX gives an error that it cant minimize the complex affine objective function, also constraints side cant be complex. I need some help to reformulate the problem.

If h^T x is complex, then CVX is absolutely doing the right thing. Minimizing a complex expression makes no sense, nor is the use of a complex expression in an inequality. You need to think carefully about what it is you are actually trying to accomplish here. Once you do that, I suspect the model will be obvious.