I am trying to find optimal solution of the following function: $$\min[\log(x^2+2y^2-xy+1)+x^2+2y^2-xy]$$
The partial derivatives are:
$$df/x=(2x-y)/(x^2-xy+2y^2+1)+2x-y$$ and $$df/y=(x-4y)/(x^2-xy+2y^2+1)-x+4y$$
The issue is that if I solve the above, I get one equation with no real solution, as pointed over here: Method for solving the following equation?
How can one approach the above problem?
We are essentially minimizing $g(z)=\log(z+1)+z$ for $z \geq 0$, since $x^2 + 2y^2 - xy = (x-0.5y)^2 + \frac{7}{4}y^2 \geq 0$. The minimum of $g$ is attained for $z=0$. In your original variables, you have $(x-0.5y)^2 + \frac{7}{4}y^2 = 0$, which only true if $x=y=0$.