I come across an optimization problem of the following form.


$$\max {\frac{1+v}{1-u}} \qquad \text{s.t.} \qquad ux^2+vy^2-xy \ge 0, \quad \forall x,y\in\mathbb{R}$$


I do not know much of optimization. What I have done is that $ux^2+vy^2\ge 2\sqrt{uv}xy\ge xy$, so I let $uv=\frac{1}{4}$ and get the seemingly correct answer.


What kind of opt problem is it? Where can I get some resources to quickly learning to solving this kind of problem?



What kind of optimization problem is this?
Let us simplify your constraints. Namely, set $x=r\sin\phi$, $y=r\cos\phi$. Then it simplifies to $u\sin^2\phi + v\cos^2\phi \geq \sin\phi \cos\phi$, for all $0\leq \phi\leq 2\pi.$ Then divide both sides by $\cos^2\phi$, and set $z:=\frac{\sin\phi}{\cos\phi}$. You'll get quadratic inequality $f(z):=uz^2-z+v\geq 0$, for all $z$. 


So we have to describe the set of $u$ and $v$ such that the latter always holds.
The discriminant of $f(z)$ is $1-4uv$, and so one has $1-4uv\leq 0$, otherwise
$f(z)$ has two distinct real roots, and $f$ can't be nonnegative everywhere.
Note that also we see, by setting $x$ or $y$ to 0, that $u\geq 0$ and $v\geq 0$.


So we simplified your constraints, getting rid of $x$ and $y$, to the following form: $uv\geq 1/4$, $u\geq 0$, $v\geq 0$. The rest looks like a standard exercise in "elementary" nonlinear optimization. 




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as suggested by the comment by Noah below, an easier way is to directly specify the constraint is to write down $$ux^2+vy^2-xy=(x,y)\begin{pmatrix}u &-1/2\&bsol;-1/2 & v\end{pmatrix}\begin{pmatrix} x\&bsol;y\end{pmatrix}$$ and note that it is nonnegative everywhere iff
$\begin{pmatrix}u &-1/2\&bsol;-1/2 & v\end{pmatrix}$ is positive semidefinite.


