I have an underdetermined system of equations of the form
\begin{equation} Ax = b, \end{equation} where $A \in \mathbf{R}^{m \times n}$ with $m < n$, subject to
\begin{equation}0 \preceq x \preceq c.\end{equation}
- I would like to know if there is any way to express the feasible set for this problem analytically.
- Is there any way to obtain any of feasible solutions in closed form?
In general, the answer is no to both questions.
Of course, you could always try a finite number of test points, including $x=0$, $x=c$, to see if they happen to satisfy $Ax=b$; and you can try the minimum-norm solution $x=A^T(AA^T)^{-1}b$, to see if it happens to satisfy $0\preceq x\preceq c$. If any of these tests hold, then you've found a closed form solution.
But again, in general, you will not be able to. It's a very simple convex optimization problem to solve numerically, though. In fact, it is a linear program, unless you choose a nonlinear function (like the norm of $x$) as an objective function. But there's no reason to do that; I'd just minimize $\sum_i x_i$ in this case, if I didn't have any other preference. Technically, your objective could just be "0" as well, in which case the point it selects will truly be solver dependent.