If you make the change of variables x(i)=u(i)-v(i) with linear constraints
and modify your least squares objective from norm(L(u-v)-R)^2 to
norm(L(u-v)-R)^2 + C*( norm(u)^2 + norm(v)^2)
then for a sufficiently small choice of C>0, this should give an equivalent solution. It's not ideal, since it forces you to re-solve with multiple choices of C, but on the other hand, it allows you to pose this as a convex problem.