With the help of the variational characterization of the zero-norm function, we reformulate the group zero-norm regularized problem as a MPCC (mathematical program with a complementarity constraint) and show that the penalty problem, yielded by moving the complementarity constraint into the objective, is a global exact penalty of the MPCC problem itself. The objective function of the exact penalty problem is not only global Lipschitz continuous in the feasible set but also has the desired bilinear structure, thereby providing a favorable equivalent Lipschitz optimization model for designing sequential convex relaxation algorithms of the group zero-norm regularized problem.