I am quite new to the field of convex optimization and in a research paper that I have read, some author represent this constraint $\frac{{{x^2}}}{{\sqrt y }} \le t$ as equivalent SOC constraint like this without explanation
Could you kindly explain to me why this is possible and how can they come up with an excellent use of the auxiliary variable $v$ to turn the original constrain into SOC form ?
Thank you for your enthusiasm !
That's because you can use SOC to write a constraint $$pq\geq r^2,$$ namely as $$\|2r,p-q\|_2\leq p+q,$$ and that makes it possible to write various simple inequalities between monomials, and then put them together to get more complicated monomials.
Since you want to model $t\sqrt{y}\geq x^2$ it feels natural to start with $$tv\geq x^2$$ and to complete the model you are forced to take $$\sqrt{y}\geq v,$$ which is also fine because it is equivalent to the same type of SOC: $$1\cdot y\geq v^2.$$
These are rather well-known tricks you can read about in https://docs.mosek.com/modeling-cookbook/cqo.html#simple-sets-involving-power-functions.
PS. The second inequality from your post looks wrong, $v$ and $y$ should be switched.