Has this differential calculus inequality approach to optimizing the production possibility curve exist? I just started micro-economics at my community college and my teacher mentioned the derivative of the PPF for two output resources. I thought about it a while and came up with this approach. Some of the notation may be incorrect as the highest math course I completed was college algebra (I tested out of it but still have todo trigonometry).

Does this approach I came up with already exist?

Here is the equations:
I don't really understand your optimization problem as it stands. Just as a few examples, when you write out

$$\min \sum_m \sum_v \left( \frac{\partial \vec f_v (\vec Q_v)}{\partial \vec f_m (\vec Q_m)} - \frac{\partial \vec f_m (\vec Q_m)}{\partial \vec f_m (\vec Q_m)} \right)$$

I have questions about why you are minimizing the difference between optimal conditions for output that has all equal opportunity costs. The justification is unclear to me. The function $\vec f_v(\vec Q_v)$ also leaves me confused looking at your work. Why is $f$ taking a vector of $n$ aggregate outputs as its argument? $f(q)$ is defined as the adjusted cost of purchasing $q \cdot n$ resources, so how does $f$ transform a vector as opposed to a scalar and what interpretation are you trying to give it?

Or in your optimization problem when you write out

$$\frac{dy}{d\sqrt{r^2 - x^2}} \rightarrow - \frac{x}{\sqrt{r^2 - x^2}} = y$$

I have no idea what you are differentiating with respect to and how you arrived at the equality on the right. In step c.) later on you substitute $x = 0$ which doesn't have a justification either. (If it does, I think you should try to be more explicit about what is happening.)



I think as is, the method of finding an optimal point on the production possibilities frontier is cumbersome, and I don't think the optimization problem is well defined.

If you wanted to refine your method, you'd have to give


conditions under where an optimum exists (the conditions you state are not sufficient)
identification strategies for an optimum
a situation where this method is more practical than the industry standard


So for that last point, why can't someone just have a production function (where they can derive the PPF) and a budget constraint from prices, regularity conditions and then solve? Since you seem to have some familiarity with higher level math, you may find Chapter 5 and 15 of Mas-Colell, Whinston, and Greene a good resources for seeing the foundation of how production problems are approached. Chapter 15 in particular more explicitly talks about production possibility sets.