Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies,
$$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} h(aX_1+bX_2+Z_1)-h(aX_1+Z_1|X_2)$$ Note that $X_1$ and $X_2$ are not independent and so we can't simplify to the following easier optimization problem $$g=\max_{} h(Y)-h(Z_1)$$ where g can be easily shown to be an upperbound on f and easily shown to admit a jointly gaussian distribution as optimal solution.
My question is can I say that since g is an upperbound on $f$, then jointly gaussian is also optimal for the optimization problem $f$?