How can I write this kind of constraint (cube over square) in cvx

2

Re-write 2^x as exp(log(2)*x)

Then use the exponential cone construct:
y*exp(y/x) <= z, with y > 0
can be written in CVX as
{ x, y, z } == exponential(1)

or alternatively,
x + rel_entr(y,z) <= 0

Declare z as a CVX variable, replace y*exp(y/x) by z, and add one of the above constraints.

Either way, CVXQUAD can be used without modification to your program, and is recommended for your problem CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions .