I have a set of measurements $M$ for a physical value $S$, that depends on 4 parameters ($a_1$, $a_2$, $b$ and $c$). There exists an approximation formula for S, but only considers 3 parameters ($a$, $b$ and $c$). Nevertheless, it is possible to get good results by considering the mean value of $a_1$ and $a_2$ and replace $a = \frac{a_1+a_2}{2}$.
I would like to explore if there is an even better mean value, utilizing the generalized mean $(\frac{1}{2} (a_1^p+a_2^p))^\frac{1}{p}$. In other words, I would like to optimize $p$ so that the sum of the squared errors (SSE) between measurements of S and their estimations from the formula is minimized.
Does anyone has a preferred tool for this job?