Product of a binary and a complex variable transformation

Nonconvex
1

Hello to the cvx community.

We know that if x is a {0,1} integer variable and y, a continuous one (with 0 \leq y \leq K) then if the product xy appears in a problem, we could replace it with a new variable z and the following set of (linear) inequalities: z\geq0\, z\leq y, z\leq Kx, y-z \leq K(1-x).

So, i wonder for a problem including complex variables, is there a similar procedure which will involve the norms of the variables? For example if v such a complex variable and w=vx the auxiliary one, I suppose that we could apply \left\| v \right\| \leq Kx with no harm, however I am confused about the applicability of the last inequality and if such an approach is valid.

Thanks to all,

a.

2

I gave it a try and came up with a possible approach: We could handle the particular complex variable v as a semi-continuous one, being 0 or belonging to the interval [a,b] where [a,b] the lower and upper bounds of \left\| v \right\|. Then ya \leq \left\| v \right\| \leq yb.

a.

3

Honestly I do not think the complex case will work. Lower bounds on complex absolute value are neither complex nor are they MIDCP-representable.