 $\omega \left(\gamma \left(\pi _{S} \right)\right)<\omega ^{*} \left(\pi _{S} \right)$ as $$\label{ZEqnNum000000}  \omega \left(\gamma \left(\pi _{S} \right)\right)\ge \omega ^{*} \left(\gamma \left(\pi _{S} \right)=0\right)-f\left(\gamma \left(\pi _{S} \right)\right)