CVX accepts your problem. The successive approximation method is failing (although with sedumi, it reports infeasible).
Even with CVXQUAD, some more work is required to avoid the successive approximation method. Use of -rel_entr(1,x) in place of log(x) and use of the explicit exponential cone construct (see mcg’s answer at Solve optimization problems of exp function ).
I think you want sum(exp(y),1) <= exp(1) , not sum(exp(y),1) <= exp(log(1)) , but doing this still results in failure with sdpt3 and infeasible with sedumi. I don;t know what will happen when you make the changes described in the previous paragraph, thereby avoiding use of the successive approximation method.