No, presuming I am correctly interpreting what your objective function is supposed to be. Please tell me if i didn’t get this correct.
Take a simple case, K = 2, and each w is a real scalar. Set \sigma^2 = 1.
Then the objective function = log2(1+w_1^2/(1+w_2^2))+log2(1+w_2^2/(1+w_1^2))
The Hessian of the objective function at w_1 = w_2 = 1 has one negative eigenvalue and one positive eigenvalue, and therefore is indefinite. So the objective function is neither concave nor convex,.
Throwing in \alpha's and binary variables as additional multiplicative factors cannot improve matters with regard to the objective function being neither concave nor convex. So CVX can not be used for this problem. if you figure out how to do so and are under 40, I will recommend you for the Fields medal.