Egyptian fractions are the representation of a rational number (for example, $\frac{61}{66}$) as a sum of reciprocals of integers (with that example, $2^{-1} + 3^{-1} + 11^{-1}$), a.k.a. unit fractions ($\frac{1}{2} + \frac{1}{3} + \frac{1}{11}$). Typically, there is an additional restriction that each integer be unique, so $\frac{2}{3}$ would be properly represented as $2^{-1} + 6^{-1}$ but not as $3^{-1} + 3^{-1}$.
For any rational number $\frac{p}{q}$, there exist infinitely many Egyptian fractions of the form $a_1^{-1} + a_2^{-1} + a_3^{-1} + ... + a_n^{-1}$.
I am interested in finding the optimal Egyptian fraction forms for a given rational number, with optimal being defined as the lowest possible sum of the constituent integers ($a_1 + a_2 + a_3 + ... + a_n$). I do not require the restriction that each integer be unique. Thus, by this metric, $3^{-1} + 3^{-1}$ would be better than $2^{-1} + 6^{-1}$ as a representation of $\frac{2}{3}$, because the value of $3 + 3$ is less than the value of $2 + 6$.
As another example, the number $\frac{13}{12}$ can be represented as $1^{-1} + 12^{-1}$ and as $2^{-1} + 3^{-1} + 4^{-1}$, with the latter representation being more-optimal than the former.
Is there a way to prove that a given Egyptian fraction is optimal, or is not optimal, given this definition?
We do have an upper bound for the sum of the constituent integers. For the rational number $\frac{p}{q}$, an optimal Egyptian fraction form must sum to less than or equal to $p × q$, because (without a restriction that all integers must be unique) we can represent $\frac{p}{q}$ as $q^{-1}$ added to itself $p$ times. So for any number $\frac{p}{q}$ and Egyptian fraction form $a_1^{-1} + a_2^{-1} + a_3^{-1} + ... + a_n^{-1}$, if the sum $a_1 + a_2 + a_3 + ... + a_n$ is greater than $p × q$ we know for sure that it is not optimal. However, there are many rational numbers with Egyptian fraction forms that are better than $p × q$, so this on its own isn't proof of optimalness.