  This paper proposes a mechanism to produce equivalent Lipschitz surrogates
for zero-norm and rank optimization problems by means of the global exact
penalty for their equivalent mathematical programs with an equilibrium
constraint (MPECs). Specifically, we reformulate these combinatorial problems
as equivalent MPECs by the variational characterization of the zero-norm and
rank function, show that their penalized problems, yielded by moving the
equilibrium constraint into the objective, are the global exact penalization,
and obtain the equivalent Lipschitz surrogates by eliminating the dual variable
in the global exact penalty. These surrogates, including the popular SCAD
function in statistics, are also difference of two convex functions (D.C.) if
the function and constraint set involved in zero-norm and rank optimization
problems are convex. We illustrate an application by designing a multi-stage
convex relaxation approach to the rank plus zero-norm regularized minimization
problem.
