Solving linear program with 1 quadratic constraint complexity
Based on posts from or-exchange and some internet reading, the following algorithm works. We adapt the cutting plane method to binary search for variable $y$. For my problem, the upper and lower bounds for all variables are known (but if these are not known, one can find a bound on $y$ by solving the LP by removing the first constraint). Let $l \leq y \leq u$.

Fix $y = (l+u)/2$. This converts the program to a linear program. If the resulting constraints are feasible, update $u(l) = (l+u)/2$. Keep performing the binary search until we reach the optimal solution with an additive $\epsilon_0$. The running time of the algorithm is $O(PTIME)\log\frac{u-l}{\epsilon_0}$. 

While this algorithm can find the optimal solution within a small additive constant, I am still not sure about the hardness of the problem for solving exactly. Any comments regarding complexity are welcome!