  Conformally K\"{a}hler, Einstein-Maxwell metrics and $f$-extremal metrics are
generalization of canonical metrics in K\"{a}hler geometry. We introduce
uniform K-stability for toric K\"{a}hler manifolds, and show that uniform
K-stability is necessary condition for the existence of $f$-extremal metrics on
toric manifolds. Furthermore, we show that uniform K-stability is equivalent to
properness of relative K-energy.
