  In this paper, we study the semi-stable twisted holomorphic vector bundles
over compact Gauduchon manifolds. By using Uhlenbeck--Yau's continuity method,
we show that the existence of approximate Hermitian--Einstein structure and the
semi-stability of twisted holomorphic vector bundles are equivalent over
compact Gauduchon manifolds. As its application, we show that the Bogomolov
type inequality is also valid for a semi-stable twisted holomorphic vector
bundle.
