I am a post graduate electrical engineering student who is working with some optimization. Suddenly, I realize that my problem ${\left( P \right)}$ is quite separable in both objective function and constraint. Particularly, my problem has the form
$\begin{array}{*{20}{c}} {\left( P \right)}&{\mathop {\min }\limits_{x,y,z} }&{{g_1}\left( x \right) + {g_2}\left( y \right) + {g_3}\left( z \right)}\\ {}&{s.t}&{{x_L} \le {f_1}\left( x \right) \le {x_U}}\\ {}&{}&{{y_L} \le {f_2}\left( y \right) \le {y_U}}\\ {}&{}&{{z_L} \le {f_3}\left( z \right) \le {z_U}} \end{array}$
Here $x_L$,$y_L$ and $z_L$ are some numerical lower bound. Likewise $x_U$,$y_U$ and $z_U$ are some numerical upper bound.
Therefore, my question is that can I find the optimal decision variable $x,y,z$ of problem $(P)$ by solving these 3 separates univariate problem $P_1$, $P_2$ and $P_3$ ?
$\begin{array}{*{20}{c}} {\left( {{P_1}} \right)}&{\mathop {\min }\limits_x }&{{g_1}\left( x \right)}\\ {}&{s.t}&{{x_L} \le {f_1}\left( x \right) \le {x_U}} \end{array}$
$\begin{array}{*{20}{c}} {\left( {{P_2}} \right)}&{\mathop {\min }\limits_y }&{{g_2}\left( y \right)}\\ {}&{s.t}&{{y_L} \le {f_2}\left( y \right) \le {y_U}} \end{array}$
$\begin{array}{*{20}{c}} {\left( {{P_3}} \right)}&{\mathop {\min }\limits_z }&{{g_3}\left( z \right)}\\ {}&{s.t}&{{z_L} \le {f_3}\left( z \right) \le {z_U}} \end{array}$
Also, if the answer is 'yes' then what would be the rigorous explanation for this approach ?
Note that sometimes $f(.)$ and $g(.)$ are not differentiable.
Thank you for your enthusiasm !