$$\max_{C_1,C_2,C_3} \frac{2C_1+C_2-\sqrt{C_2^2+4C_1C_3}}{2(C_1+C_2-C_3)}:=G(C_1,C_2,C_3)$$
s.t. $$C_1\in(0,2],C_2\in(0,1],C_3\in(0,2]$$ $$D\in(0,1]$$ $$C^2D^4-2BCD^3>0 \Longleftrightarrow C_1+C_2-C_3<0$$ $$\frac{C_1^2+C_3^2}{2}D-D^2\frac{C_1^2+C_3^2}{2}-4C_2^2D^4-6D^2C_2^2+2C_2^2D+8C_2^2D^3>0$$ $$C_1\leq C_3$$
(From Mathematica this region look is equivalent to s.t. $$C_1\in(0,2],C_2\in(0,1],C_3\in(0,2]$$ $$C_1+C_2-C_3<0$$(but I haven't proved yet)
where $$B=2C_1+C_2$$ $$C:=C_1+C_2-C_3$$ and $$D:=\frac{2C_1+C_2-\sqrt{C_2^2+4C_1C_3}}{2(C_1+C_2-C_3)}\text{ i.e., the objective function.}$$
I am confused on how to deal with the singular points where $C_1+C_2-C_3=0$.