I'm a software engineer and have not much mathematical knowledge. Now, I'm facing with a problem in my research. I have a system of equations as below: $$P_1 = \alpha V_p + \beta I_c^2 $$ $$P_2 = \alpha V_c + \beta I_l^2 $$ $$ I_c = (P_p - P_1)/V_c$$ $$ I_l = (V_c * I_c - P_2)/V_b$$ in these equations $ \alpha, \beta, V_p, P_p$ and $V_b$ are constants. I want minimize $P_1 + P_2$ by choosing an optimal value for $V_c$. thanks in advance
I am afraid that an analytical solution could be a monster (if any). Considering $$P_1 = \alpha V_p + \beta I_c^2\tag 1$$ $$P_2 = \alpha V_c + \beta I_l^2\tag 2$$ $$I_c = (P_p - P_1)/V_c\tag 3$$ $$I_l = (V_c * I_c - P_2)/V_b\tag 4$$ you could replace $I_c$ from $(3)$ and $I_l$ fom$(4)$ by their expressions in $(1)$ and $(2)$. So, you have two quite complex polynomial equations in $P_1$ and $P_2$. Yoy could solve one of them (say $(2)$) to get $P_2$ as a function of $P_1$; however, the problem is which root to select ?
Assume you know; so now you want to minimize $P_1+P_2(P_1)$ with respect to $V_c$; this makes another monster.
If I had to do it, I should consider numerical methods for optimization under four equality constraints $(1,2,3,4)$ and I suppose some bound constraints such as $P_1>0$, $P_2>0$ (if they apply).
If you have a set of parameters to provide, I could try nd show the results. I do not think that the problem is difficult from a numerical point of view.