I need to minimize a sum of the form
$$ \arg\min_{y_{1}, y_{2}, y_{3}} \sum_{i} \left( x_{i,1} - \frac {a y_{1} + b y_{3} + x_{i,2} y_{3}} {c y_{2} + b y_{3} + x_{i,3} y_{3}} \right)^2 $$
Where $y_{1}, y_{2}, y_{3}$ are the unknown parameters, and $x_{i,1}, x_{i,2}, x_{i,3}$ are the known data values, different for each $i$. $a, b, c, d$ are known constants.
Is there a way to transform this into a linear least squares problem, and solve it using a closed-form expression?
Let us assume that the model is $$ x_{i,1} = \frac {a y_{1} + b y_{3} + x_{i,2} y_{3}} {c y_{2} + b y_{3} + x_{i,3} y_{3}}$$ It is highly nonlinear and you need reasonable guesses to start the nonlinear regression. So, in a first step, consider that the model is $$x_{i,1} (c y_{2} + b y_{3} + x_{i,3} y_{3})=a y_{1} + b y_{3} + x_{i,2} y_{3}$$ and define the residues as $$r_i=(cx_{1,i})y_2+(bx_{1,i}+bx_{1,i}x_{3,i}-b-x_{2,i})y_3-a y_1$$ This is then equivalent to a linear regression problem since you will minimize $$\text{SSQ}=\sum_{i=1}^n r_i^2$$ When solved, you can start the nonlinear regression.