I would like to obtain a least squares solution for an under-determined system of linear equations,
$$Ax=b$$
where, $A \in \mathbb{R}^{7\times9}, \, x \in \mathbb{R}^{9\times1}\text{and } b\in\mathbb{R}^{7\times1}$.
Clearly, there is no unique solution, but I would like to obtain a solution that satisfies some non-linear inequality conditions for a few components in $x$,
\begin{align} f_1(x_1,x_2,x_3) &\le c_1 \\ f_2(x_7,x_8,x_9) &\le c_2 \\ \end{align}
If it helps, the starting point $x_\text{init}$ is known apriori (but I am happy to discard this knowledge if so needed).
I am happy with any solution $x^*$ that minimises $\|Ax -b\|_2$ whilst satisfying the two constraints above.
I am currently using MATLAB as my programming environment, but pseudo-code/links to useful resources etc shall be more than enough to get started. An example MATLAB code for a dummy problem shall also be very helpful.