Consider the following linear system \begin{equation} y = A_1 x_1 + A_2 x_2 \end{equation} subject to the linear constrains \begin{equation} C_1 x_1 + C_2 x_2 \leq d \end{equation}
I am looking for a solution for the above linear system that gives more priority to the coordinates corresponding to $x_1$ compared to those of $x_2$. This is what I precisely mean by priority
If $x_1$ alone can geenerated the given $y$ while $x_2$ is kept at its minimum feasible value.
If there are infinite number of solutions for $x_1$ in step 1 then we take the minimum norm solution.
If $x_1$ alone cannot generate $y$ then we allow $x_2$ to participate in generating $y$ with the optimal $x_1$ obtained in step 1 and 2.
If there are infinite solutions for $x_2$, then we take the minimum norm solution.
How to formuate the above described problem as an optimization problem for instance QP?
PS. I am not interested in weighted solutions for the sake of priority. Priority is required to be strict.
I would try to describe my problem in another way. But maximizing $x_1$ probably will not be the best solution. I need $x_1$ to minimize $(y-A_1 x_1)^T(y-A_1 x_1)$. If we can find $x_1$ that allows this cost to be zero, then there is no need for $x_2$ at all and it will be set to zero of couse if $0 \in {Cx \leq d}$ (let's take it as an assumption for the moment). Probably, there may a finite number of solutions for $x_1$ that let the cost $(y-A_1 x_1)^T(y-A_1 x_1)$ to be zero. In this case, I would seek the solution that minimises $x_1^Tx_1$. If there is a residual from the above problem then we let $x_2$ to minimize this residual i.e. minimise $(y-A_1 x_1^* - A_2 x_2 )^T(y-A_1 x_1^* - A_2 x_2)$ subject to $ C_2 x_2 \leq d-C_1 x_1^*$
I'm not sure I fully understand your problem, but I think what you want is to solve $$ \max_{x_1,x_2}\;x_1 $$ subject to \begin{cases} A_1x_1+A_2x_2=y\\ C_1x_1+C_2x_2\le d \end{cases}
This is straightforward with the simplex algorithm.