Presuming g and w are real column vectors, the LHS is a real scalar, but the constraint is non-convex. it can be handled in CVX by using Big M logic modeling for non-convex use of abs, as shown in the y = ans(x) section of of Logics and integer-programming representations - YALMIP .
You will need a solver which can handle binary (or integer) variables. Specifically Mosek,. If there are no SDP constraints, Gurobi is another solver option, although there may be Gurobi incompatibilities with CVX nowadays).
Thank you for your help. Additionally, what should I do if w and g are complex vectors？
I’ll leave it to others to address whether there is an MIDCP formulation which covers that.
OK, thank you very much.