I have a single objective optimization problem. The problem can consist of $n$ independent variables $(x_1,x_2,...,x_n)$ four functions $f_i(\bf{x})$ and a function $Q(\bf{x})$.
I want to monitor how a composite function $F$, composed of four individual functions $f_1(x_1,x_2\ldots x_n),f_2(x_1,x_2,\ldots,x_n),f_3(x_1,x_2\ldots,x_n),f_4(x_1,x_2,\ldots,x_n)$, tracks a the rate of change (gradient) of the function $Q(x_1,x_2,\ldots,x_n)$ for $x_n\in[\alpha,\beta]$ with $\alpha,\beta\in\mathbb{R}$.
Ι will probably use the weighted sum method to compose $F$ so $F$ will be in the form $F=a_1f_1+a_2f_2+a_3f_3+a_4f_4$ with $\displaystyle \sum_ia_i=1$ to sweep through different combinations of weights.
My question is how would I compare the rates of change between $F$ and $Q$? If I had a single independent variable, I would probably create $J = Q'-F'$ and I would try to minimize it.
I thought something along the lines of $J = \nabla Q -\nabla F$ but would this hold?
Is there any general approach in these kinds of problems?