Hello. I am working on an optimization problem that involves writing a least-squares minimization problem. I would like to know if my steps are correct or if I misinterpreted the question. I'll post the question and my work below and will award the best response a medal.

Question

anonymous · Answer

Assume that m points $$(t_i, b_i)$$ are given in the plane i=1,...,m. Formulate a least squares minimization problem in the unknown $$x=(x_1,x_2,x_3,x_4,x_5)^T$$, for fitting a curve defined by$$b(t)=x_1+x_2e^{x_3t}+x_4e^{x_5t}$$ through the points in an optimal way.


My attempt at the solution is: $$Minimize_{x_1,x_2,x_3,x_4,x_5} f(x)=\sum_{1}^{m}r_i(x)^2$$ where m=5(because there are only 5 x variables) and $$r_i(x)=b_i-(x_1+x_2e^{x_3t_i}+x_4e^{x_5t_i})$$
Afterwards I just wrote down $$r_1(x),...,r_5(x)$$ My textbook also mentions that I can write this as $$f(x)=r^T(x)r(x)$$, but I decided to do this using sigma notation. What I am not sure about is if I approached this the correct way, if I'm missing something, or if I showed enough steps. Thank you.