I have the following problem
Let $x\in\mathbb{R}^{6\times1}$ be unknown decision variables, $b\in\mathbb{R}^{3 \times 1}$ is a known vector, and $A$ is a known matrix. We formulate the following optimization problem
\begin{array}{lc} \text{minimize} & \Big\lVert \left[x_1, x_2,x_3\right]^\text{T} \Big\rVert_2 \\ \text{subject to} \\ & \det R > 0\\ & A x = b \\ \text{where} \\ & R(x), A\in\mathbb{R}^{3\times6} \end{array}
The solution process includes iterating $x_1,x_2,x_3$ while $x_4,x_5,x_6$ are uniquely determined for each iteration of $x_1,x_2,x_3$. Then $x_4,x_5,x_6$ seem to be something like intermediate variables. Is this the proper way to formulate the optimization problem with respect to $x_4,x_5,x_6$? They can be any values as long at the constraints are met. But, as soon as we choose $x_1,x_2,x_3$ then $x_4,x_5,x_6$ are uniquely determined.
