I have a system of convex constraints of the form:
$|\vec{k} + \vec{p}_1| + |\vec{k} + \vec{p}_2| + |\vec{k} - \vec{l} + \vec{p}_3| \leq o_4$
for variables $\vec{k}$ and $\vec{l}$ and constants $\vec{p}_i$ and $o_i$.
I am looking for an efficient way to determine whether this system has a solution in $\vec{k}$ and $\vec{l}$. I am not interested in determining an actual solution.
My current workaround is to use the system as constraints in a second-order cone program where I minimize the function 1. This seems like overkill, so I am wondering if there is a quicker way to determine if the set of feasible points is not empty than performing a minimization procedure.
If the minimization procedure is indeed the fastest way, is it safe to set the desired accuracy very low?