I am struggling with the following optimization problems.
Problem 1
\begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 +c_1 + s_1) + s_2 (K_2+ c_2 + s_2) + 2\alpha K) +C\\\text{s.t.}&\quad s_1 \geq 0, s_2 \geq 0, \alpha \geq 0, \alpha \geq A-s_1 - \beta s_2\end{align}
Problem 2
\begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 + s_1) + s_2 (K_2 + s_2) + \alpha K)\\\text{s.t.}&\quad s_1 \geq 0, s_2 \geq 0, \alpha \geq 0, B-s_1 - \beta s_2 \leq \alpha \leq A-s_1 - \beta s_2\end{align}
where $K_1 >0, K_2 >0, C>0, 0<\beta<1, 0<B<A, \gamma >0, c_1, c_2 >0 $ are constants.
If for a given $\gamma = \gamma'$, the optimal objective value of Problem 1 is greater than that of Problem 2, is the optimal objective value of Problem 1 greater than that of Problem 2 for all $0< \gamma < \gamma'$?
I can prove this when the constraints $s_1, s_2, \alpha \geq 0$ do not exist. I proved it by determining a closed form solution for each problem and just compared the two (derivative with respect to $\gamma$ yields lower value for Problem 1 for all $\gamma >0$). How can this be solved when the non-negativity constraints are introduced?