Unfeasible linear program becomes feasible if a variable is removed
There is (unless $P=NP$) no polynomial time algorithm for this problem, since the problem is $NP$-hard by reduction from Subset Sum. If you set $l_i=u_i$ then the problem is to determine if there is a subset of the $u_i$ that sums to $C$ (which is the Subset Sum problem).

If your problem did not allow $l_i=u_i$, if you were to require that $l_i&lt;x_i&lt;u_i$, you could get around this by setting $10000\cdot l_i&lt;x_i&lt;10000\cdot u_i + 1$ which for sufficiently large values of $10000$ ensures that $x_i$ will be "close enough".