Consider the following quadratic program
$$\begin{array}{ll} \underset{x,y}{\text{minimize}} & a x^2 - b x y - c y\\ \text{subject to} & x \geq 1\\ & y \geq 5\end{array}$$
where $a$, $b$ and $c$ are given scalars. This problem arises as I was modelling a transistor's minimal dimensions $x$ and $y$ for it to operate in the saturation region and behave as an audio amplifier.
Unfortunately, I don't see this quadratic objective as a convex. Therefore, I have no idea what might be a good approach for this problem. What I did essentially is assume $x = 1$ and compute $y$ directly.
Update: The goal of this question perhaps was to generalize the expression that I have ended up facing:
Let $x$ and $y$ be the length and width of the transistor accordingly, I seek to design a transistor that has a minimum length of $1$ $\mu\text{m}$ and minimum width of $5$ $\mu\text{m}$. The equations that govern the behavior of the transistor has landed me in the following equation : $$ \theta(x,y)=1.5x^{2}-xy-0.05y $$ The goal is to find optimal parameters $x$ and $y$ that minimizes $\theta$ while obeying these constraints. Graphically, and for other values of $a$, $b$, and $c$ there doesn't appear to be a convex form for the objective function.