  We propose a new description of the SU(N) Yang-Mills theory on a lattice,
which enables one to explain quark confinement based on the dual
superconductivity picture in a gauge independent way. This is because we can
define gauge-invariant magnetic monopoles which are inherent in the Wilson loop
operator. For SU(3) there are two options: the minimal option with a single
type of non-Abelian magnetic monopole characterized by the maximal stability
subgroup $\tilde{H}=U(2)=SU(2)\times U(1)$, and the maximal one with two types
of Abelian magnetic monopoles characterized by the maximal torus subgroup
$\tilde{H}=U(1)\times U(1)$. The maximal option corresponds to a gauge
independent reformulation of the Abelian projection represented by the
conventional MAG. In the minimal option, we have successfully performed the
numerical simulation of the SU(3) Yang-Mills theory on a lattice. We give
preliminary numerical results showing the dominance of the non-Abelian magnetic
monopole in the string tension obtained from the Wilson loop in the fundamental
representation, and the infrared dominance of a decomposed field variable for
correlation functions after demonstrating the preservation of color symmetry
which was explicitly broken by the conventional MAG.
