I’m afraid we’re basically back to where we were on your previous thread I have a constraint, that I am unable to write in CVX acceptable syntax. Help! .
Consider the special case in which all MATLAB variables other than those declared to be variables in CVX have the value 1. Then ignoring the index i for simplicity of exposition, your first constraint can be put in the form 1/w-log(1+p/w) <= 0. Even though you have declared p to be a CVX variable, if we fix p for the moment at 1, then this is exactly the same constraint which was analyzed in the aforementioned thread, and it is convex, but per mcg’s answer there, it is (apparently) not representable in CVX. Unfortunately, when we allow p to be a non-negative CVX variable, 1/w-log(1+p/w) <= 0 need not even be convex, as evidenced by the 2nd derivative of the LHS w.r.t w being -0.0055 at p=10, w=10.
As a minor footnote, your parentheses placement w.r.t. S^2 is inconsistent between your model statement and your CVX code, but that does not affect my conclusions above, since I set S equal to 1.